CONTENTS

I Łukasiewicz Logic and MV-Algebras I Antonio Di Nola and Ioana Leus¸tean 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 MV-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The lattice structure of an MV-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Examples of MV-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The interval MV-algebra of an `u-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The distance function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Ideals, filters, and homomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 The Boolean center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 The Riesz decomposition property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Ideals in MV-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The lattice of the ideals of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Congruences and quotient MV-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Maximal ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 The spectral topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Classes of MV-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 MV-chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Simple and semisimple MV-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Local and perfect MV-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 MV-algebras and Abelian `u-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The functor Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Properties of Γ(G, u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Chang’s completeness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Functional representation of semisimple MV-algebras . . . . . . . . . . . . . . . . . 5.5 Perfect MV-algebras and Abelian `-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Representations by ultrapowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Łukasiewicz ∞-valued logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The syntax of Ł . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Lindenbaum–Tarski algebra (Ł(Θ), Ł) . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 4 6 11 15 17 19 20 24 25 27 28 30 33 35 36 39 44 44 47 49 58 58 66 70 73 75 79 81 81 87

2

6.3 Consistent sets, linear sets, and deductive systems . . . . . . . . . . . . . . . . . . . . 88 6.4 The semantics of Ł . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3

7 Varieties of MV-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.1 Komori’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.2 Equational characterization of varieties of MV-algebras . . . . . . . . . . . . . . . 107 8 Historical remarks and further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Chapter I: Łukasiewicz Logic and MV-Algebras A NTONIO D I N OLA AND I OANA L EUS¸ TEAN

1

Introduction

MV-algebras were defined by Chang [12] as algebraic structures corresponding to Łukasiewicz ∞-valued propositional logic Š [50]. This logical system has the real unit interval [0, 1] as set of truth values and its basic logical connectives, ∗ (negation) and → (implication), can be expressed using the real algebraic operations: a∗ = 1 − a and

a → b = min{1, 1 − a + b}.

The usual lattice operations on [0, 1] (a ∨ b = max{a, b}, a ∧ b = min{a, b}) are defined by a ∨ b = (a → b) → b and a ∧ b = (a∗ ∨ b∗ )∗ . The negation ∗ is an involution, but the excluded middle principle is not satisfied. If one defines on [0, 1] the operations ⊕ and by a ⊕ b = min{a + b, 1}

and

a b = (a∗ ⊕ b∗ )∗ = max{a + b − 1, 0},

then a ⊕ a∗ = 1 and a a∗ = 0. Note that ⊕ is in fact the real numbers addition truncated to [0, 1] and is its dual with respect to involutive negation. Moreover, the lattice operations can be also expressed by: a ∨ b = a ⊕ (b a∗ ) = b ⊕ (a b∗ ) and

a ∧ b = a (b ⊕ a∗ ) = b (a ⊕ b∗ ).

When he defined the structure of MV-algebra, Chang’s main goal was to provide an algebraic proof for the completeness theorem of Łukasiewicz propositional calculus Š (i.e., any formula of Š is provable if and only if it holds in [0, 1]) [13]. He chose ⊕, and ∗ as primary operations. Many of his initial axioms are inspired by the properties that are satisfied on [0, 1] [12]. A main tool in Chang’s proof of the completeness theorem was the bijective correspondence between the linearly ordered MV-algebras and the linearly ordered Abelian lattice ordered groups with strong unit. The deep connection between MV-algebras and Abelian lattice ordered groups with strong unit was completely clarified by Mundici in [60]: the two categories are equivalent. Many important aspects in the theory of MV-algebras have their origin in this correspondence. The historical remarks at the end of this chapter aim to give an account on the various developments in MV-algebras, as well as suggestions for further reading. A standard reference for the elementary theory of MV-algebras is [18], while [71] approaches advanced topics.

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Antonio Di Nola and Ioana Leus¸tean

This chapter presents some fundamental aspects of MV-algebra theory and it is intended to be self-contained. Some results are particular cases of results proved in other chapters. Some definitions and constructions are just specializations of general notions in logic and universal algebra. We included them in our presentation for the sake of completeness. Łukasiewicz logic, developed in Section 6 is a core fuzzy logic, so main theorems are instances of more general results from Chapters I and II. Note that Chapter III contains a proof theoretic approach to Łukasiewicz logic, while its computational complexity is analyzed in Chapter X. Since MV-algebras are BL-algebras with involutive negation, aspects of MV-algebra theory directly follow from Chapter V. The second section of Chapter IX presents two important issues concerning MV-algebras: McNaughton theorem and functional characterization of the free structures.

2

Definitions and basic properties

We give below a simplified definition of MV-algebras, which is due to Mangani [52]. We prove that the class of MV-algebras is polynomially equivalent with the class of Wajsberg algebras, structures that are obtained through a direct algebraization of Š [33]. We provide few examples of MV-algebras, some of which will be frequently referred. We prove that an interval of an Abelian lattice ordered group can be endowed with an MV-algebra structure. The fundamental connection between MV-algebras and Abelian lattice ordered groups will be deeply analyzed in Section 5. 2.1

MV-algebras

DEFINITION 2.1.1. An MV-algebra is a structure hA, ⊕, ∗ , 0i, where ⊕ is a binary operation, ∗ is a unary operation and 0 is a constant such that the following axioms are satisfied for any a, b ∈ A: (MV1) hA, ⊕, 0i is an Abelian monoid, (MV2)

(a∗ )∗ = a,

(MV3)

0∗ ⊕ a = 0∗ ,

(MV4)

(a∗ ⊕ b)∗ ⊕ b = (b∗ ⊕ a)∗ ⊕ a.

In order to simplify the notation, an MV-algebra hA, ⊕, ∗ , 0i will be referred by its support set, A. An MV-algebra is trivial if its support is a singleton. On an MValgebra A we define the constant 1 and the auxiliary operation as follows: 1 := 0∗

a b := (a∗ ⊕ b∗ )∗

for any a, b ∈ A. We will also use the notation a∗∗ := (a∗ )∗ . Unless otherwise specified by parentheses, the order of evaluation of these operations is first ∗ , then , and finally ⊕. Using these notations, the axioms (MV3) and (MV4) have the equivalent forms (MV30 ) and respectively (MV40 ): (MV30 ) 0

(MV4 )

1 ⊕ a = 1, (a b∗ ) ⊕ b = (b a∗ ) ⊕ a,

for any a, b ∈ A. A list of direct consequences is given below.

Chapter I: Łukasiewicz Logic and MV-Algebras

7

PROPOSITION 2.1.2. If A is an MV-algebra then the following identities hold for any a, b and c ∈ A: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

1∗ = 0, a ⊕ a∗ = 1, a 1 = a, a 0 = 0, a a∗ = 0, (a b)∗ = a∗ ⊕ b∗ , (a ⊕ b)∗ = a∗ b∗ , a (a∗ ⊕ b) = b (b∗ ⊕ a), a (b c) = (a b) c, a b = b a.

PROPOSITION 2.1.3. If A is an MV-algebra and a ∈ A, then x = a∗ is the unique solution of the system a ⊕ x = 1, a x = 0. Proof. If x is the solution of the above system, then x∗ a∗ = (x ⊕ a)∗ = 1∗ = 0 by Proposition 2.1.2 (g). It follows that x = x ⊕ 0 = x ⊕ (x∗ a∗ ). Using (MV40 ) and (MV2) we get x = a∗ ⊕ (a x) = a∗ ⊕ 0 = a∗ . PROPOSITION 2.1.4. In any MV-algebra A there is at most one element a ∈ A such that a = a∗ . Proof. Let us suppose that there are a, b ∈ A such that a = a∗ and b = b∗ . Using (MV30 ), (MV40 ) and Proposition 2.1.2 (c), (b), (h), (g) we obtain: a = a 1 = a (a∗ ⊕a⊕b) = (a⊕b) (a∗ b∗ ⊕a) = (a⊕b) (a∗ b⊕a) = (a⊕b) (b∗ a⊕b) = (a ⊕ b) (b∗ a∗ ⊕ b) = b (b∗ ⊕ a ⊕ b) = b 1 = b. Hence, if there is an element a ∈ A such that a = a∗ , then a is the unique element satisfying this property. NOTATION 2.1.5. Let A be an MV-algebra, a ∈ A and n ∈ N. We introduce the following notations: 0a = 0, a0 = 1,

na = a ⊕ (n − 1)a an = a (an−1 )

for any n ≥ 1, for any n ≥ 1.

We say that the element a has order n, and we write ord (a) = n, if n is the least natural number such that na = 1. We say that the element a has a finite order, and we write ord (a) < ∞, if a has order n for some n ∈ N. If no such n exists, we say that a has infinite order and we write ord (a) = ∞. DEFINITION 2.1.6. Let hA, ⊕, ∗ , 0i be an MV-algebra and B ⊆ A such that the following conditions are satisfied: (S1) 0 ∈ B, (S2)

if a, b ∈ B, then a ⊕ b ∈ B,

(S3)

if a ∈ B, then a∗ ∈ B.

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Antonio Di Nola and Ioana Leus¸tean

Thus, if we consider the restriction of ⊕ and ∗ to B, we get an MV-algebra hB, ⊕, ∗ , 0i which is an MV-subalgebra (or, simply, subalgebra) of the MV-algebra A. If S ⊆ A is a subset of A, then we will denote by hSi the least subalgebra of A which includes S and it will be called the subalgebra generated by S in A. We will say that S is a system of generators for hSi. EXAMPLE 2.1.7. In any MV-algebra A the subalgebra generated by the empty set is h∅i = {0, 1}. 2.2

The lattice structure of an MV-algebra In the sequel, A will be an MV-algebra.

PROPOSITION 2.2.1. For any a, b ∈ A the following are equivalent: (a)

a∗ ⊕ b = 1,

(b)

a b∗ = 0,

(c) b = a ⊕ b a∗ , (d)

there is c ∈ A such that b = a ⊕ c,

(e)

there is d ∈ A such that a = b d. ∗

∗

Proof. (a) ⇒ (b) a b∗ = (a∗ ⊕ b∗∗ ) = (a∗ ⊕ b) = 1∗ = 0. (b) ⇒ (c) a ⊕ b a∗ = b ⊕ a b∗ = b ⊕ 0 = b. (c) ⇒ (d) We consider c = b a∗ . (d) ⇒ (e) For d = a ⊕ b∗ we get b d = a (b ⊕ a∗ ) = a (a ⊕ c ⊕ a∗ ) = a 1 = a. (e) ⇒ (a) a∗ ⊕ b = b∗ ⊕ d∗ ⊕ b = 1 ⊕ d∗ = 1. DEFINITION 2.2.2. We define a binary relation ≤ on A by a ≤ b iff a and b satisfy one of the equivalent conditions from Proposition 2.2.1. PROPOSITION 2.2.3. ≤ is an order relation on A. Proof. The relation ≤ is reflexive, since a ⊕ a∗ = 1. If a ≤ b and b ≤ a, then b = a ⊕ b a∗ and a = b ⊕ a b∗ . Using the axiom (MV40 ) we get a = b, so ≤ is an antisymmetric relation. In order to prove the transitivity, we consider a ≤ b and b ≤ c. Thus, there are x and y ∈ A such that b = a ⊕ x and c = b ⊕ y. Hence c = a ⊕ (x ⊕ y), so a ≤ c. PROPOSITION 2.2.4. For any a, b and c ∈ A, the following properties hold: (a)

0 ≤ a ≤ 1,

(b)

a ≤ b iff b∗ ≤ a∗ ,

(c) a b ≤ c iff a ≤ b∗ ⊕ c, (d)

a ≤ b implies a ⊕ c ≤ b ⊕ c and a c ≤ b c,

(e) a c∗ ≤ a b∗ ⊕ b c∗ , (f) a (b ⊕ c) ≤ b ⊕ (a c).

Chapter I: Łukasiewicz Logic and MV-Algebras

9

Proof. Let a, b and c be arbitrary elements of A. (a) follows by (MV30 ) and Proposition 2.1.2 (d). (b) (c)

a ≤ b iff a∗ ⊕ b = 1 iff a∗ ⊕ b∗∗ = 1 iff b∗ ≤ a∗ . ∗ a b ≤ c iff (a b) ⊕ c = 1 iff a∗ ⊕ b∗ ⊕ c = 1 iff a ≤ b∗ ⊕ c. ∗

(d) We get b⊕c⊕(a ⊕ c) = b⊕(c⊕a∗ c∗ ) = b⊕(a∗ ⊕a c) = (b⊕a∗ )⊕a c = 1 ⊕ a c = 1, so a ⊕ c ≤ b ⊕ c. The other inequality follows similarly. (e) By (c), it suffices to prove that a ≤ c⊕a b∗ ⊕b c∗ . We have c⊕a b∗ ⊕b c∗ = (c⊕b c∗ )⊕a b∗ = (b⊕c b∗ )⊕a b∗ = (b⊕a b∗ )⊕c b∗ = a⊕b a∗ ⊕c b∗ . Thus, the desired inequality becomes a ≤ a ⊕ b a∗ ⊕ c b∗ , which obviously holds by Definition 2.2.2 and Proposition 2.2.1 (a). ( f ) We show that (a (b ⊕ c))∗ ⊕ (b ⊕ (a c)) = 1. Indeed we have: (a (b ⊕ c))∗ ⊕ (b⊕(a c)) = a∗ ⊕(b⊕c)∗ ⊕b⊕(a c) = (a∗ ∨c)⊕b⊕(b∗ c∗ ) = (a∗ ∨c)⊕(b∨c∗ ) = 1. Hence, by Proposition 2.2.1 (a) we get the desired inequality. We introduce two auxiliary operations, ∨ and ∧, by setting a ∨ b := a ⊕ b a∗ = b ⊕ a b∗

and

a ∧ b := a (b ⊕ a∗ ) = b (a ⊕ b∗ ).

PROPOSITION 2.2.5. If a, b ∈ A, then: (a)

a b ≤ a ∧ b ≤ a ≤ a ∨ b ≤ a ⊕ b,

(b)

(a ∨ b) = a∗ ∧ b∗ ,

∗

∗

(c) (a ∧ b) = a∗ ∨ b∗ , (d)

a ⊕ b = (a ∨ b) ⊕ (a ∧ b),

(e) a b = (a ∨ b) (a ∧ b), (f) a ∧ b = 0 implies a ⊕ b = a ∨ b, (g)

a ∨ b = 1 implies a b = a ∧ b.

Proof. (a) By definition it is obvious that a ∧ b ≤ a ≤ a ∨ b. In order to prove the ∗ other two inequalities, we will use Proposition 2.2.1 (a). Thus, (a ∧ b) ⊕ (a b) = a (a∗ ⊕ b) ⊕ a∗ ⊕ b∗ = (a (a∗ ⊕ b) ⊕ a∗ ) ⊕ b∗ = (a∗ ⊕ b ⊕ (a∗ a b)) ⊕ b∗ = 1, ∗ and also (a ⊕ b) ⊕ (a ∨ b) = b ⊕ (a ⊕ a∗ (a ⊕ b∗ )) = b ⊕ a ⊕ b∗ ⊕ (a a∗ b) = 1. ∗

∗

(b)

(a ∨ b) = (a ⊕ a∗ b) = a∗ (a ⊕ b∗ ) = a∗ (a∗∗ ⊕ b∗ ) = a∗ ∧ b∗ .

(c)

(a ∧ b) = (a (a∗ ⊕ b)) = a∗ ⊕ a b∗ = a∗ ⊕ a∗∗ b∗ = a∗ ∨ b∗ .

(d)

(a ∨ b) ⊕ (a ∧ b) = a ⊕ (a∗ b) ⊕ (b (a ⊕ b∗ )) = a ⊕ (a∗ b) ⊕ b (a∗ b) = a ⊕ b ⊕ (b∗ a∗ b) = a ⊕ b ⊕ 0 = a ⊕ b.

(e)

a b = a∗ ⊕ b∗ ∗ = ((a∗ ∨ b∗ ) ⊕ (a∗ ∧ b∗ )) = (a∗ ∨ b∗ ) (a∗ ∧ b∗ ) = (a∗∗ ∧ b∗∗ ) (a∗∗ ∨ b∗∗ ) = (a ∧ b) (a ∨ b).

(f)

Follows by (d).

(g)

Follows by (e).

∗

∗

∗

∗

∗

∗

10

Antonio Di Nola and Ioana Leus¸tean

PROPOSITION 2.2.6. The partially ordered set hA, ≤i is a bounded lattice such that 0 is the first element, 1 is the last element and l .u.b.{a, b} = a ∨ b

g.l .b.{a, b} = a ∧ b,

for any a, b ∈ A. Proof. By Proposition 2.2.4 (a), it follows that 0 is the first element and 1 is the last element of A. In order to prove that l .u.b.{a, b} = a∨b, note that a ≤ a∨b and b ≤ a∨b by Proposition 2.2.5 (a). If c ∈ A such that a ≤ c and b ≤ c, then c = a ⊕ a∗ c and ∗ c⊕b∗ = 1. We get c⊕(a ∨ b) = a⊕a∗ c⊕(a∗ ∧b∗ ) = a∗ c⊕(a⊕a∗ (a⊕b∗ )) = a∗ c ⊕ a ⊕ b∗ ⊕ a a∗ b = (a∗ c ⊕ a) ⊕ b∗ ⊕ 0 = a c∗ ⊕ c ⊕ b∗ = a c∗ ⊕ 1 = 1, so a ∨ b ≤ c. Now we prove that g.l .b.{a, b} = a ∧ b. By Proposition 2.2.5 (a), we have a∧b ≤ a and a∧b ≤ b. Let c be in A such that c ≤ a and c ≤ b. By Proposition 2.2.4 (b), it follows that a∗ ≤ c∗ and b∗ ≤ c∗ . Hence l .u.b.{a∗ , b∗ } = a∗ ∨ b∗ ≤ c∗ . Using again ∗ Proposition 2.2.4 (b), we get c ≤ (a∗ ∨ b∗ ) , so c ≤ a ∧ b by Proposition 2.2.5 (c). We will denote L(A) = hA, ∨, ∧, 0, 1i, the lattice structure of A. We call L(A) the lattice reduct of A. DEFINITION 2.2.7. An MV-algebra A is complete (σ-complete) if the lattice reduct of A is a complete (σ- complete) lattice. W from A, then we will write ai instead of W If {ai | i ∈ I}Vis a family of elements V {ai | i ∈ I} and ai instead of {ai | i ∈ I} if there are no possible confusions. LEMMA 2.2.8. Let {ai | i ∈ I} be a family of elements from A. W V V W ∗ (a) If ai exists, then ai ∗ exists and ai ∗ = ( ai ) . V W W V ∗ (b) If ai exists, then ai ∗ exists and ai ∗ = ( ai ) . Proof. We will W use Proposition 2.2.4 (b) and Proposition 2.2.5 (b), (c). (a) If a = ai , then ai ≤ a for any i ∈ I. It follows that ai ∗ ≥ a∗ for any i ∈ I, so a∗ is a lower bound of the family {ai ∗ | i ∈ I}. Let z ∈ A such that z ≤ ai ∗ for any i ∈ I. We have z ∗ ≥ ai for any i ∈ I, so a ≤ z ∗ . Thus, z ≤ a∗ , so a∗ is the greatest lower bound of the family {ai ∗ | i ∈ I} in A. (b) follows similarly. PROPOSITION 2.2.9. For any a ∈ A and W for any family of Velements {bi | i ∈ I} ⊆ A, the following properties hold whenever {bi | i ∈ I} and {bi | i ∈ I} exist: W W (a) a {bi | i ∈ I} = {a bi | i ∈ I}, W W (b) a ∧ {bi | i ∈ I} = {a ∧ bi | i ∈ I}, W W (c) a ⊕ {bi | i ∈ I} = {a ⊕ bi | i ∈ I}, V V (d) a ⊕ {bi | i ∈ I} = {a ⊕ bi | i ∈ I}, V V (e) a ∨ {bi | i ∈ I} = {a ∨ bi | i ∈ I}, V V (f) a {bi | i ∈ I} = {a bi | i ∈ I}.

Chapter I: Łukasiewicz Logic and MV-Algebras

11

W Proof. (a) It is obvious that a bi ≤ a bi for any i ∈ I. Let z be another upper bound of the family {a bi | i ∈ I}, so a bi ≤ z for any i ∈ I. We get bi ≤ bi ∨ a∗ = a∗ ⊕ a bi ≤ a∗ ⊕ z W W for any i ∈ I. It follows that bi ≤ a∗ ⊕ z. Thus, we have a bi ≤ a (a∗ ⊕ z) = W a ∧ z ≤ z, so a Wbi is the least upper bound of the family {a bi | i ∈ I}. (b) We have a ∧ bi ≥ aW ∧ bi for any i ∈ I. We consider z ∈ A such that z ≥ a ∧ bi ∗ for any i ∈ I. Since bi ∗ ≥ ( bi ) , we have  _ ∗  z ≥ a ∧ bi = bi (a ⊕ bi ∗ ) ≥ bi a ⊕ bi , for any i ∈ I. Using (a), it follows that _   _ ∗  _   _ ∗  _ a ∧ bi = bi a ⊕ bi a ⊕ bi = bi ≤ z. W Thus, a ∧ bi is theWleast upper bound of the family {a ∧ bi |Wi ∈ I}. (c) We have a ⊕ bi ≥ a ⊕ bi for any i ∈ I. Hence, a ⊕ bi is an upper bound of the family {a ⊕ bi | i ∈ I}. Let z be another upper bound, so z ≥ a ⊕ bi for any i ∈ I. We get z ≥ a and z a∗ ≥ (a ⊕ bi ) a∗ = a∗ ∧ bi for any i ∈ I. Using (b) it follows that _ _ a∗ ∧ bi = (a∗ ∧ bi ) ≤ z a∗ . We infer that a⊕

_

 _  bi = a ⊕ a∗ ∧ bi ≤ a ⊕ z a∗ = a ∨ z = z,

W so a ⊕ bi is the least upper bound of the family {a ⊕ bi | i ∈ I}. (d), (e) and (f) Follows from (a), (b) and (c), respectively, using Lemma 2.2.8. We will only give the proof of (d):  ∗ ^ ^ _ ∗ _ ^ a ⊕ bi = a∗ bi ∗ = (a∗ bi ∗ ) = (a∗ bi ∗ )∗ = (a ⊕ bi ). PROPOSITION 2.2.10. If a, x1 , . . . , xn ∈ A for some n ≥ 1, then the following properties hold: (a)

a ∨ (x1 ⊕ · · · ⊕ xn ) ≤ (a ∨ x1 ) ⊕ · · · ⊕ (a ∨ xn ),

(b)

a ∧ (x1 ⊕ · · · ⊕ xn ) ≤ (a ∧ x1 ) ⊕ · · · ⊕ (a ∧ xn ),

(c) a ∨ (x1 · · · xn ) ≥ (a ∨ x1 ) · · · (a ∨ xn ), (d)

a ∧ (x1 · · · xn ) ≥ (a ∧ x1 ) · · · (a ∧ xn ).

Proof. (a)

We prove by induction on n ≥ 1. For n = 2 we use Proposition 2.2.9 (c):

(a ∨ x1 ) ⊕ (a ∨ x2 ) = (a ⊕ a) ∨ (a ⊕ x2 ) ∨ (a ⊕ x1 ) ∨ (x1 ⊕ x2 ) ≥ a ∨ a ∨ a ∨ (x1 ⊕ x2 ) = a ∨ (x1 ⊕ x2 ).

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Now, the induction step follows: a ∨ (x1 ⊕ · · · ⊕ xn+1 ) ≤ (a ∨ (x1 ⊕ · · · ⊕ xn )) ⊕ (a ∨ xn+1 ) ≤ (a ∨ (x1 ) ⊕ · · · ⊕ (a ∨ xn ) ⊕ (a ∨ xn+1 ). (b), (c) and (d) follows similarly. PROPOSITION 2.2.11. For any a, b ∈ A the following properties hold: (a)

(a b∗ ) ∧ (a∗ b) = 0,

(b)

(a ⊕ b∗ ) ∨ (a∗ ⊕ b) = 1,

(c) a ∧ a∗ ≤ b ∨ b∗ , (d)

a ⊕ b = a ⊕ b ⊕ a b,

(e) a ⊕ b ≤ 2a ∨ 2b. Proof. (a) 0 = (a∧b) (a∧b)∗ = (a∧b) (a∗ ∨b∗ ) = (a (a∗ ∨b∗ ))∧(b (a∗ ∨b∗ )) = (a a∗ ∨ a b∗ ) ∧ (b a∗ ∨ b b∗ ) = (0 ∨ a b∗ ) ∧ (b a∗ ∨ 0) = (a b∗ ) ∧ (b a∗ ). (b)

Using (a) we get 1 = 0∗ = ((a b∗ ) ∧ (b a∗ ))∗ = (a∗ ⊕ b) ∨ (b∗ ⊕ a).

(c) We will use (a) and Proposition 2.2.1 (b). Thus, we have (a ∧ a∗ ) (b ∨ b∗ )∗ = (a ∧ a∗ ) (b ∧ b∗ ) = (a b) ∧ 0 ∧ (a∗ b∗ ) = 0, and so a ∧ a∗ ≤ b ∨ b∗ . (d) From (b) we get 1 = (a ⊕ b) ∨ (a∗ ⊕ b∗ ) = (a ⊕ b) ⊕ (a ⊕ b)∗ (a∗ ⊕ b∗ ) = (a ⊕ b) ⊕ a∗ b∗ (a b)∗ = (a ⊕ b) ⊕ (a ⊕ b ⊕ a b)∗ . Using Proposition 2.2.1 (a), it follows that a ⊕ b ≥ a ⊕ b ⊕ a b. Since the converse inequality is also true by Proposition 2.2.5 (a), the desired result holds. (e) By Proposition 2.2.5 (d) we have a ⊕ b = (a ∨ b) ⊕ (a ∧ b) = (a ⊕ (a ∧ b)) ∨ (b ⊕ (a ∧ b)) = (2a ∧ (a ⊕ b)) ∨ (2b ∧ (a ⊕ b)) = (a ⊕ b) ∧ (2a ∨ 2b) ≤ 2a ∨ 2b. PROPOSITION 2.2.12. Let a and b be two elements of A. (a)

If a ∧ b = 0, then na ∧ nb = 0 for any n ∈ N.

(b)

If a ∨ b = 1, then an ∨ bn = 1 for any n ∈ N.

Proof. (a) If a∧b = 0 we have a = a⊕(a∧b) = 2a∧(a⊕b) ≥ 2a∧b. It follows that 2a ∧ b ≤ a ∧ b = 0, so 2a ∧ b = 0. Moreover, b = b ⊕ (2a ∧ b) = (b ⊕ 2a) ∧ 2b ≥ 2a ∧ 2b, so 2a ∧ 2b ≤ 2a ∧ b = 0. Thus we proved that a ∧ b = 0 implies 2a ∧ 2b = 0. By induction, it follows that 2k a ∧ 2k b = 0 for any k ∈ N. For any n ∈ N there is k ∈ N such that n ≤ 2k . Hence na ∧ nb ≤ 2k a ∧ 2k b = 0, so na ∧ nb = 0. (b) If a ∨ b = 1, then a∗ ∧ b∗ = 0. Using (a), we get na∗ ∧ nb∗ = 0 for any n ∈ N. It follows that 1 = 0∗ = (na∗ ∧ nb∗ )∗ = an ∨ bn for any n ∈ N. COROLLARY 2.2.13. For any a, b ∈ A and for any n ∈ N we have: (a)

n(a b∗ ) ∧ n(b a∗ ) = 0,

(b)

(a ⊕ b∗ )n ∨ (b ⊕ a∗ )n = 1.

Proof. (a) Follows by Proposition 2.2.11 (a) and Proposition 2.2.12 (a). (b) Follows by Proposition 2.2.11 (b) and Proposition 2.2.12 (b).

Chapter I: Łukasiewicz Logic and MV-Algebras

13

COROLLARY 2.2.14. For any a, b ∈ A, if ord (a b) < ∞, then a ⊕ b = 1. Proof. By hypothesis, for some n ≥ 1, n(a b) = 1. Hence, by Corollary 2.2.13 (a), we get n(b∗ a∗ ) = 0. It follows that (a ⊕ b)n = 1, so a ⊕ b = 1. We recall that a Kleene algebra is a structure hL, ∨, ∧, ∗ , 0, 1i where hL, ∨, ∧, 0, 1i is a bounded distributive lattice and ∗ is a unary operation such that the following properties hold for any a, b ∈ L: (K1)

(a∗ )∗ = a,

(K2)

(a ∨ b)∗ = a∗ ∧ b∗ ,

(K3)

a ∧ a∗ ≤ b ∨ b∗ .

PROPOSITION 2.2.15. For any MV-algebra A, the structure hL(A), ∨, ∧, ∗ , 0, 1i is a Kleene algebra. Proof. By Proposition 2.2.6, the structure hL(A), ∨, ∧, 0, 1i is a bounded lattice. The lattice L(A) is distributive (Proposition 2.2.9 (b) and (e)). The property (K1) is (MV2), (K2) follows from Proposition 2.2.5 (b) and (K3) follows by Proposition 2.2.11 (c). 2.3

The implication

In the sequel, hA, ⊕, , ∨, ∧, 0, 1i will be an MV-algebra. The implication is defined by a → b := a∗ ⊕ b for any a, b ∈ A. LEMMA 2.3.1. The following properties hold for any a, b ∈ A: (a)

a∗ = a → 0,

(b)

a ≤ b iff a → b = 1,

(c) a ∧ b = a (a → b), (d)

a ∨ b = (a → b) → b,

(e) (a → b)n ∨ (b → a)n = 1 for any n ∈ N. Proof. (a), (b), (c) follows by definition. (d) (a → b) → b = (a → b)∗ ⊕ b = (a∗ ⊕ b)∗ ⊕ b = (a b∗ ) ⊕ b = a ∨ b, (e) is straightforward by Corollary 2.2.13 (b). LEMMA 2.3.2. For any a ∈ A andWfor any family of V elements {bi | i ∈ I} ⊆ A, the following properties hold whenever {bi | i ∈ I} and {bi | i ∈ I} exist: W W (a) a → {bi | i ∈ I} = {a → bi | i ∈ I}, V V (b) a → {bi | i ∈ I} = {a → bi | i ∈ I}, W V (c) ( {bi | i ∈ I}) → a = {bi → a | i ∈ I}, V W (d) ( {bi | i ∈ I}) → a = {bi → a | i ∈ I}. Proof. We will only prove (a) and (c), using Proposition 2.2.9 and De Morgan laws.

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W W W ∗ ∗ (a) a → W {bi | i ∈ I} = a ⊕ {bi | i ∈ I} = {a ⊕ bi | i ∈ I} = = {a → bi | i ∈ I}. W W V ∗ (c) ( V {bi | i ∈ I}) → a = ( V{bi | i ∈ I}) ⊕ a = ( {bi ∗ | i ∈ I}) ⊕ a = = {bi ∗ ⊕ a | i ∈ I} = {bi → a | i ∈ I}. LEMMA 2.3.3. If a, b, x ∈ A, then: (a)

a x ≤ b iff x ≤ a → b,

(b)

a → b = max{x ∈ A | a x ≤ b}.

Proof. (a) follows by Proposition 2.2.4 (c). (b) is a direct consequence of (a). COROLLARY 2.3.4. The structure hL(A), ∨, ∧, , →, 0, 1i is a residuated lattice for any MV-algebra A. PROPOSITION 2.3.5. In any MV-algebra A the following properties hold: (W1)

1 → a = a,

(W2)

(a → b) → ((b → c) → (a → c)) = 1,

(W3)

(a → b) → b = (b → a) → a,

(W4)

(a∗ → b∗ ) → (b → a) = 1.

Proof. (W1) 1 → a = 0 ⊕ a = a. (W2) Using consecutively Lemma 2.3.1 (b), Lemma 2.3.3 (a) and Proposition 2.2.1 (b), it follows that (a → b) → ((b → c) → (a → c)) = 1 iff (a → b) ≤ (b → c) → (a → c) iff (a → b) (b → c) ≤ a → c iff (a → b) (b → c) (a → c)∗ = 0. We prove the last equality: (a → b) (b → c) (a → c)∗ = (a → b) (b → c) a c∗ = a (a∗ ⊕ b) c∗ (b∗ ⊕ c) = (a ∧ b) (c∗ ∧ b∗ ) ≤ b b∗ = 0, and so (a → b) (b → c) (a → c)∗ = 0. (W3) By Proposition 2.3.1 (d), (a → b) → b = a ∨ b = b ∨ a = (b → a) → a. (W4) (a∗ → b∗ ) → (b → a) = (a⊕b∗ )∗ ⊕b∗ ⊕a = a∗ b⊕b∗ ⊕a = (a∗ ∨b∗ )⊕a = (a∗ ⊕ a) ∨ (b∗ ⊕ a) = 1 ∨ (b∗ ⊕ a) = 1. DEFINITION 2.3.6. A Wajsberg algebra is a structure hW, →, ∗, 1i, where → is a binary operation, ∗ is a unary operation, and 1 is a constant and the identities (W1)– (W4) from Proposition 2.3.5 hold. COROLLARY 2.3.7. If hA, ⊕, ∗ , 0i is an MV-algebra, then WA = hA, →, ∗ , 1i is a Wajsberg algebra, where → is the MV-algebra implication and 1 = 0∗ . In the sequel, we will prove that the converse is also true, i.e. any Wajsberg algebra has an MV-algebra structure. Moreover, the category of MV-algebras and the category of Wajsberg algebras are equivalent.

Chapter I: Łukasiewicz Logic and MV-Algebras

15

PROPOSITION 2.3.8 ([33]). In any Wajsberg algebra hW, →, ∗ , 1i the following properties hold for any a, b and c ∈ W : (a)

a → a = 1,

(b)

if a → b = 1 and b → a = 1, then a = b,

(c)

if a → b = 1 and b → c = 1, then a → c = 1,

(d)

a → 1 = 1,

(e) a → (b → a) = 1, (f) a → ((a → b) → b) = a → ((b → a) → a) = 1, (g)

a → (b → c) = b → (a → c),

(h)

(a → b) → ((c → a) → (c → b)) = 1,

(i) 1∗ → a = 1, (j) a = a∗ → 1∗ , (k) a∗ = a → 1∗ , ∗

(l) (a∗ ) = a, (m) a∗ → b∗ = b → a. Proof. (a) Using (W2) and (W1) we get (1 → 1) → ((1 → a) → (1 → a)) = 1, 1 → (a → a) = 1, and so finally a → a = 1. (b) By (W1) and (W3) it follows that a = 1 → a = (b → a) → a = (a → b) → b = 1 → b = b. (c) Straightforward by (W2) and (W1). (d) From (W3) and (a) we infer that (a → 1) → 1 = (1 → a) → a = a → a = 1. Thus, by (W2) and (W1), it follows that (1 → a) → ((a → 1) → (1 → 1)) a → ((a → 1) → 1) a→1 (e)

= 1 = 1 = 1.

Using (W2), (W1) and (d) we get (b → 1) → ((1 → a) → (b → a)) = 1 → (a → (b → a)) = a → (b → a) =

1 1 1.

( f ) By (e), a → ((b → a) → a) = 1. The other equality follows by (W3). (g) If we denote x = ((b → c) → c) → (a → c) = ((c → b) → b) → (a → c) then (a → (b → c)) → x = 1 by (W2). Moreover, using also (f) we get (b → ((b → c) → c)) → (x → (b → (a → c))) 1 → (x → (b → (a → c))) (x → (b → (a → c)))

= 1 = 1 = 1.

By (c), we have (a → (b → c)) → (b → (a → c)) = 1 for any a, b ∈ W . Hence, we also have (b → (a → c)) → (a → (b → c) = 1, so the desired equality follows by (b).

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(h) Straightforward by (W2) and (g). ( i ) Using (W1) and (W4), we get (a∗ → 1∗ ) → a = (a∗ → 1∗ ) → (1 → a) = 1. Thus, by (h) we obtain ((a∗ → 1∗ ) → a) → ((1∗ → (a∗ → 1∗ )) → (1∗ → a)) = 1 1 → ((1∗ → (a∗ → 1∗ )) → (1∗ → a)) = 1 (1∗ → (a∗ → 1∗ )) → (1∗ → a) = 1. From (e) and (W1), we get 1∗ → a = 1. ( j ) By (e) and (W4), we have a∗ → (1∗∗ → a∗ ) = 1 and (1∗∗ → a∗ ) → (a → 1∗ ) = 1. Using (c), it follows that a∗ → (a → 1∗ ) = 1 so, by (g), a → (a∗ → 1∗ ) = 1. By (W4), we also have (a∗ → 1∗ ) → a = (a∗ → 1∗ ) → (1 → a) = 1. Hence, from (b), we deduce that a = a∗ → 1∗ . (k) By (W1), (i), (W3) and (j) we obtain that: a∗ = 1 → a∗ = (1∗ → a∗ ) → a∗ = (a∗ → 1∗ ) → 1∗ = a → 1∗ . ∗ ( l ) By (k) and (j) we get: (a∗ ) = a∗ → 1∗ = a. ∗ ∗ (m) By (l) and (W4) we get (b → a) → (a∗ → b∗ ) = ((b∗ ) → (a∗ ) ) → (a∗ → b∗ ) = 1. The desired equality follows by (W4) and (b). FACT 2.3.9. If hW, →, ∗ , 1i is a Wajsberg algebra and we define a ≤ b iff a → b = 1, for any a, b ∈ W , then ≤ is a partial order on W by Proposition 2.3.8 (a), (b), (c). By (d) and (i), it follows that 1 is the last element and 1∗ is the first element. Moreover, one can prove that hW, ≤i is a lattice where for any a, b ∈ A: a ∨ b = (a → b) → b and

a ∧ b = (a∗ ∨ b∗ )∗ .

PROPOSITION 2.3.10. If in a Wajsberg algebra hW, →, ∗ , 1i we define a ⊕ b = a∗ → b and

0 = 1∗

for any a, b ∈ W , then AW = hW, ⊕, ∗ , 0i is an MV-algebra. Proof. (MV1) We have to prove that hW, ⊕, 0i is an Abelian monoid. The commutativity of ⊕ is straightforward using Proposition 2.3.8 (m) and (l): ∗

a ⊕ b = a∗ → b = b∗ → (a∗ ) = b∗ → a = b ⊕ a. From (j), we get a ⊕ 0 = a∗ → 1∗ = a. Now, using the commutativity of ⊕ and (g), we are able to prove the associativity: a ⊕ (b ⊕ c) = a ⊕ (c ⊕ b) = a∗ → (c∗ → b) = c∗ → (a∗ → b) = c ⊕ (a ⊕ b) = (a ⊕ b) ⊕ c. (MV2) Follows from Proposition 2.3.8 (l). (MV3) By Proposition 2.3.8 (l), we get 0∗ = 1. Thus, using (i) it follows that 0∗ ⊕ a = 1 ⊕ a = 1∗ → a = 1 = 0∗ . (MV4) Follows from (W4): (a ⊕ b∗ )∗ ⊕ a = (a ⊕ b∗ ) → a = (b → a) → a = (a → b) → b = (b ⊕ a∗ )∗ ⊕ b.

Chapter I: Łukasiewicz Logic and MV-Algebras

17

COROLLARY 2.3.11. For any MV-algebra A and for any Wajsberg algebra W , the following properties hold: (a)

AWA = A,

(b)

WA W = W .

Proof. Note that the support sets of AWA and A coincide, as well as the ∗ operation. Similarly, the support sets and the operation ∗ on WAW and W coincide. Thus, we only have to prove that ⊕AWA = ⊕A and →WAW = →W . (a) If we denote B = AWA , then for any a, b ∈ A we get a ⊕B b = a∗ →WA b = ∗ (a∗ ) ⊕A b = a ⊕A b. (b) If we denote by V = WAW , then for any a, b ∈ W we get a →V b = a∗ ⊕AW b = ∗ (a∗ ) →W b = a →W b. 2.4

Examples of MV-algebras We will describe some basic examples of MV-algebras.

EXAMPLE 2.4.1. Any Boolean algebra is an MV-algebra in which the operations ⊕ and ∨ coincide, EXAMPLE 2.4.2 ([0, 1], Z ∩ [0, 1], Łn+1 ). Let R denote the set of real numbers and let Z denote the set of rational numbers. For any n ∈ N, n ≥ 1 we define Ln+1 = {0, 1/n, . . . , (n − 1)/n, 1}. If a and b are real numbers we define a ⊕ b := min{a + b, 1}

and a∗ := 1 − a.

One can easily see that the unit interval [0, 1], the set Z ∩ [0, 1] and the set Ln+1 with n ≥ 1 are closed under the above defined operations. Straightforward computations show that h[0, 1], ⊕, ∗ , 0i, hZ ∩ [0, 1], ⊕, ∗ , 0i, and hLn+1 , ⊕, ∗ , 0i are MV-algebras, which will be simply denoted by [0, 1], Z ∩ [0, 1] and Łn+1 respectively. If n = 1, then Ln+1 = L2 = {0, 1}, the Boolean algebra with two elements. Moreover, the auxiliary operation is given by a b = max{a + b − 1, 0} and the order is the natural order of the real numbers. EXAMPLE 2.4.3 (AX , [0, 1]X ). Let hA, ⊕, ∗ , 0i be an MV-algebra and X a nonempty set. The set AX of all the functions f : X → A becomes an MV-algebra with the pointwise operations, i.e., if f , g ∈ AX , then (f ⊕g)(x) := f (x)⊕g(x), f ∗ (x) := f (x)∗ for any x ∈ X and 0 is the constant function associated with 0 ∈ A. A special significance has the MV-algebra [0, 1]X , where [0, 1] is the MV-algebra defined in Example 2.4.2. An element f ∈ [0, 1]X is called fuzzy subset of X and, for any x ∈ X, f (x) represents the degree of membership of x to f . The subalgebras of [0, 1]X are called bold algebras of fuzzy sets. EXAMPLE 2.4.4 (C(X)). Let X be a topological space and consider [0, 1] the unit real interval equipped with the natural topology. We consider C(X) = {f : X → [0, 1] | f is continuous}.

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One can easily see that C(X) is a subset of [0, 1]X from Example 2.4.3 and C(X) is closed under the MV-algebra operations defined pointwise. Thus, if f , g ∈ C(X), then f ⊕ g and f ∗ ∈ C(X) where (f ⊕ g)(x) = min{f (x) + g(x), 1} and f ∗ (x) = 1 − f (x) for any x ∈ X. We obtain the MV-algebra hC(X), ⊕, ∗ , 0i, where 0 is the constant function associated with 0 ∈ [0, 1]. EXAMPLE 2.4.5 (Chang’s MV-algebra C). Let {c, 0, 1, +, −} be a set of formal symbols. For any n ∈ N we define the following abbreviations:  if n = 0,  0 c if n = 1, nc :=  c + (n − 1)c if n > 1.  if n = 0,  1 1−c if n = 1, 1 − nc :=  1 − (n − 1)c − c if n > 1. We consider C = {nc | n ∈ N} ∪ {1 − nc | n ∈ N} and we define the MV-algebra operations as follows: (⊕1)

if x = nc and y = mc, then x ⊕ y := (m + n)c,

(⊕2)

if x = 1 − nc and y = 1 − mc, then x ⊕ y := 1,

(⊕3)

if x = nc and y = 1 − mc and m ≤ n, then x ⊕ y := 1,

(⊕4)

if x = nc and y = 1 − mc and n < m, then x ⊕ y := 1 − (m − n)c,

(⊕5)

if x = 1 − mc and y = nc and m ≤ n, then x ⊕ y := 1,

(⊕6)

if x = 1 − mc and y = nc and n < m, then x ⊕ y := 1 − (m − n)c,

∗

( 1)

if x = nc, then x∗ := 1 − nc,

(∗ 2)

if x = 1 − nc, then x∗ := nc.

Hence, the structure C = hC, ⊕, ∗ , 0i is an MV-algebra, which is called Chang’s algebra since it was defined by Chang [12]. The order relation is defined by: x≤y

iff x = nc and y = 1 − mc or x = nc and y = mc and n ≤ m or x = 1 − nc and y = 1 − mc and m ≤ n.

Since 0 ≤ c ≤ · · · ≤ nc ≤ · · · ≤ 1 − nc ≤ · · · ≤ 1 − c ≤ 1, C is an MV-chain. EXAMPLE 2.4.6 (The interval algebra A(0, a)). Let hA, ⊕, ∗ , 0i be an MV-algebra and assume a > 0 in A. Denote A(0, a) = [0, a] = {x ∈ A | 0 ≤ x ≤ a}. For any x, y ∈ A(0, a) define x ⊕[0,a] y := (x ⊕ y) ∧ a

and

x∗[0,a] := x∗ a.

The structure hA(0, a), ⊕[0,a] , ∗[0,a] , 0i is an MV-algebra.

Chapter I: Łukasiewicz Logic and MV-Algebras

19

EXAMPLE 2.4.7 (The Lindenbaum–Tarski algebra Ł). From the logical point of view, the most important example of an MV-algebra is the algebra Ł arising from the ∞-valued Łukasiewicz propositional logic, which will be developed in Section 6. The formulas in this logic are built up of denumerable many propositional variables with two operations ¬ and → inductively in the following manner: (f1) every propositional variable is a formula, (f2) if ϕ is a formula, then ¬ϕ is a formula, (f3) if ϕ and ψ are formulas, then (ϕ → ψ) is a formula, (f4) a string of symbols is a formula of Š iff it can be shown to be a formula by a finite number of applications of (f1), (f2), and (f3). We will denote by Fm Š the set of all formulas of Š. The particular four axiom schemes of this propositional calculus are: (A1) ϕ → (ψ → ϕ), (A2) (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)), (A3) ((ϕ → ψ) → ψ) → ((ψ → ϕ) → ϕ), (A4) (¬ψ → ¬ϕ) → (ϕ → ψ). The deduction rule is modus ponens (MP): from ϕ and ϕ → ψ infer ψ. The notation ` ϕ means that the formula ϕ is provable from (A1)–(A4) using only modus ponens. On the set of all formulas we define the equivalence relation ≡ by: ϕ ≡ ψ iff ` ϕ → ψ and ` ψ → ϕ. If we denote by [ϕ] the equivalence class of the formula ϕ determined by ≡, then Š = {[ϕ] | ϕ is a formula} is the set of all the equivalence classes. It is obvious that Ł is a Wajsberg algebra. Consequently, the MV-algebra operations on Ł are defined as follows: [ϕ] ⊕ [ψ] := [¬ϕ → ψ] and [ϕ]∗ := [¬ϕ]. If we also define 0 := [ϕ] for any ϕ such that ` ¬ϕ, then the structure Ł = hŠ, ⊕, ∗ , 0i is an MV-algebra. One can easily prove that [ϕ] ∨ [ψ] = [(ϕ → ψ) → ψ] and 1 = [ϕ] iff ` ϕ. 2.5

The interval MV-algebra of an `u-group

A lattice-ordered group (`-group) is a structure hG, +, 0, ≤i such that hG, +, 0i is a group, hG, ≤i is a lattice and any group translation is isotone. For a comprehensive study of `-groups theory one can see [3, 7, 8, 34]. If G and H are `-groups, then h : G → H is an `-group homomorphism if h is both a group homomorphism and a lattice homomorphism. For G an `-group we denote by G+ the positive cone of G. If g ∈ G, then the positive and, respectively, the negative part of g are g+ = g ∨ 0 and g− = (−g) ∨ 0. We remind that g = g+ − g− and |g| = g ∨ (−g) = g+ + g− . A positive element u in G is a strong unit if for every g ∈ G+ there is n ∈ N such that g ≤ nu. An Abelian `-group with strong unit hG, ui will be simply called `u-group. We begin with the simple, but very important observation that, given an Abelian `-group hG, +, 0, ≤i and a positive element u > 0 in G, the interval [0, u] can be endowed with an MV-algebra structure. Moreover, in Section 5 we will prove that any MV-algebra coincides with the interval [0, u] of an `u-group hG, ui.

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LEMMA 2.5.1. Let hG, +, 0, ≤i be an Abelian `-group and u ∈ G such that u > 0. If we define on the interval [0, u] = {a ∈ G | 0 ≤ a ≤ u} the operations a ⊕ b := (a + b) ∧ u and

a∗ := u − a,

then the structure h[0, u], ⊕, ∗ , 0i is an MV-algebra. Moreover, for any a, b ∈ [0, u] we get a b = (a + b − u) ∨ 0 and the lattice operations on [0, u] are the restriction of the lattice operations on G. Proof. We will only prove the associativity of ⊕ and the axiom (MV4), since the other axioms are straightforward. For any a, b and c ∈ [0, u], we have (a ⊕ b) ⊕ c

= (((a + b) ∧ u) + c) ∧ u = (a + b + c) ∧ (u + c) ∧ u = (a + b + c) ∧ u = (a + b + c) ∧ (a + u) ∧ u = (a + ((b + c) ∧ u) ∧ u = a ⊕ (b ⊕ c).

a ⊕ (b∗ ⊕ a)∗

= (a + (u − ((u − b + a) ∧ u))) ∧ u = (a + ((b − a) ∨ 0)) ∧ u = (b ∨ a) ∧ u = b ∨ a.

The axiom (MV4) is a direct consequence of the last equality. If G is an Abelian `-group and u > 0 in G, then the MV-algebra h[0, u], ⊕, ∗ , 0i from Lemma 2.5.1 will be denoted by [0, u]G . EXAMPLE 2.5.2. (1) If G = R, the `-group of the real numbers and u = 1, then [0, u]G is the MV-algebra [0, 1] from Example 2.4.2. (2)

If G = Z and u = n, then [0, u]G is isomorphic with the MV-algebra Łn+1 from Example 2.4.2.

EXAMPLE 2.5.3 (∗ [0, 1]). Let R be the set of the real numbers, P(N) the Boolean algebra of all the subsets of N and F ⊆ P(N) the ultrafilter which contains all the cofinite subsets (i.e., the sets with finite complements). We denote ∗ R := RN /F, the ultrapower of R in the class of `-groups. The elements of ∗ R are called nonstandard reals. We will briefly describe the structure of ∗ R. If f , g : N → R are two elements from RN we define f ∼ g iff {n ∈ N | f (n) = g(n)} ∈ F. One can easily prove that ∼ is an equivalence, so we consider ∗ R = RN /F = {[f ] | f ∈ RN } the set of all the equivalence classes with respect to ∼. If we define [f ] + [g] := [f + g] and [f ] ≤ [g] iff {n ∈ N | f (n) ≤ g(n)} ∈ F then ∗ R becomes an `-group. Moreover, since F is an ultrafilter, ∗ R is linearly ordered. A real element of ∗ R is an element of the form [r] where r is a constant function RN . An infinitesimal is an element τ ∈ ∗ R such that |τ | ≤ [1/n] for any n ∈ N, where |τ | = max{τ, −τ } is the absolute value of τ . For example, if t | N → R is defined by t(0) = 0 and t(n) = 1/n for n > 0, then τ = [t] is an infinitesimal in ∗ R. Results from nonstandard analysis shows that any nonstandard real has one of the forms [r] + τ or [r] − τ where [r] is a real and τ is an infinitesimal. Now, we consider the interval ∗ [0, 1] = {[f ] ∈ ∗ R | [0] ≤ [f ] ≤ [1]} and we define the operations [f ] ⊕ [g] := max{[f ] + [g], [1]}

and

[f ]∗ := [1] − [f ]

for any [f ], [g] ∈ ∗ [0, 1]. As in Example 2.4.2, h∗ [0, 1], ⊕, ∗ , [0]i is an MV-algebra.

Chapter I: Łukasiewicz Logic and MV-Algebras

21

A standard construction in `-group theory is the lexicographic product: if G1 is a totally ordered group and G2 is an `-group, then their lexicographic product is the `group G1 ×lex G2 , whose support set is G1 × G2 , the group operations are defined on components but the order relation is lexicographic: hx1 , x2 i ≤ hy1 , y2 i

iff

x1 < y1 or x1 = y1 and x2 ≤ y2 ,

for any x1 , y1 ∈ G1 and x2 , y2 ∈ G2 . EXAMPLE 2.5.4 (Komori chains K n ). If G = Z ×lex Z, then the MV-algebras K n+1 = [h0, 0i, hn, 0i]G where n > 0 are called Komori chains. These structures were introduced by Komori [43] and they are used for characterizing the equational classes of MV-algebras (Section 7). Note that K 2 = [h0, 0i, h1, 0i]G is isomorphic with Chang’s algebra C from Example 2.4.5. 2.6

The distance function In an MV-algebra hA, ⊕, ∗ , 0i we define the distance function d : A × A → A by d(a, b) := (a b∗ ) ⊕ (b a∗ ).

PROPOSITION 2.6.1. For any a, b, x, y ∈ A the following properties hold: (d1) (d2) (d3) (d4) (d5) (d6) (d7) (d8) (d9)

d(a, b) = a b∗ ∨ b a∗ , d(a, b) = 0 iff a = b, d(a, 0) = a, d(a, 1) = a∗ , d(a∗ , b∗ ) = d(a, b), d(a, b) = d(b, a), d(a, c) ≤ d(a, b) ⊕ d(b, c), d(a ⊕ c, b ⊕ e) ≤ d(a, b) ⊕ d(c, e), d(a c, b e) ≤ d(a, b) ⊕ d(c, e).

Proof. (d1) Follows by Propositions 2.2.5 (d) and 2.2.11 (a). (d2) If a = b, then it is obvious that d(a, b) = 0. Conversely, if d(a, b) = 0, then a b∗ = b a∗ = 0. We get a ≤ b and b ≤ a, so a = b. (d3), (d4), (d5), (d6) Follows by easy computations. (d7) Using Proposition 2.2.4 (e) we have d(a, c) = a c∗ ⊕ c a∗ ≤ (a b∗ ⊕ b c∗ ) ⊕ (c b∗ ⊕ b a∗ ) = (a b∗ ⊕ b a∗ ) ⊕ (b c∗ ⊕ c b∗ ) = d(a, b) ⊕ d(b, c). (d8) We first prove that inequality (a ⊕ c)∗ (b ⊕ e) ≤ b a∗ ⊕ e c∗ . We have ((a ⊕ c)∗ (b ⊕ e))∗ ⊕ b a∗ ⊕ e c∗ = a ⊕ c ⊕ b∗ e∗ ⊕ b a∗ ⊕ e c∗ = (a ⊕ b a∗ ) ⊕ (c ⊕ e c∗ ) ⊕ b∗ e∗ = (b ⊕ a b∗ ) ⊕ (c ∨ e) ⊕ (b∗ e∗ ) = (b ⊕ b∗ e∗ ) ⊕ (c ∨ e) ⊕ a b∗ = (b ∨ e∗ ) ⊕ (c ∨ e) ⊕ a b∗ ≥ e∗ ⊕ e = 1. The inequality then follows by Proposition 2.2.1 (a). Now we prove (d8) using the above inequality twice: d(a ⊕ c, b ⊕ e) = (a ⊕ c)∗ (b ⊕ e) ⊕ (b ⊕ e)∗ (a ⊕ c) ≤ (b a∗ ⊕e c∗ )⊕(a b∗ ⊕c e∗ ) = (b a∗ ⊕a b∗ )⊕(e c∗ ⊕c e∗ ) = d(a, b)⊕d(c, e). (d9) Follows by (d5) and (d8): d(a c, b e) = d(a∗ ⊕ c∗ , b∗ ⊕ e∗ ) ≤ d(a∗ , b∗ ) ⊕ d(c∗ , e∗ ) = d(a, b) ⊕ d(c, e).

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EXAMPLE 2.6.2. (1) If A is a Boolean algebra, then the distance function is d(a, b) = (a ∧ b∗ ) ∨ (b ∧ a∗ ) = (a ↔ b)∗ . (2)

In the MV-algebras [0, 1], Z ∩ [0, 1] and Łn from Example 2.4.2 the distance function is d(a, b) = |a − b|, where |r| denotes the absolute value of r for any real number r.

(3)

In Chang’s MV-algebra C the distance function is given by d(nc, mc) = d(1 − nc, 1 − mc) = |n − m|c, d(nc, 1 − mc) = 1 − (n + m)c for any n, m ∈ N, where |n − m| is the absolute value of n − m.

(4)

In the MV-algebra Łfrom Example 2.4.7, the distance function is d([ϕ], [ψ]) = (ϕ → ψ) → ¬(ψ → ϕ).

(5)

If A is an MV-algebra and a ∈ A such that 0 < a, then the distance function in the interval algebra A(0, a) is d[0,a] (x, y)

= (x∗[0,a] [0,a] y) ∨[0,a] (y ∗[0,a] [0,a] x) = (x∗ y) ∨ (y ∗ x) = d(x, y),

where x, y ∈ [0, a] and d(x, y) is the distance function in A. 2.7

Ideals, filters, and homomorphisms In the sequel A is an MV-algebra.

DEFINITION 2.7.1. A nonempty set I ⊆ A is an ideal if the following properties are satisfied: (I1) a ≤ b and b ∈ I implies a ∈ I, (I2) a, b ∈ I implies a ⊕ b ∈ I. We will denote by Id (A) the set of all the ideals of A. An ideal is proper W if it does not coincide withWthe entire algebra. An ideal I is closed if X ⊆ I implies {x | x ∈ X} ∈ I whenever {x | x ∈ X} exists in A. FACT 2.7.2. The following properties are straightforward: (i1) {0} and A are ideals, (i2) 0 ∈ I for any ideal I of A, (i3)

an ideal I is proper iff 1 ∈ / I,

(i4)

if I is a proper ideal and a ∈ I, then ord (a) = ∞,

(i5)

if I is an ideal and a, b ∈ I, then a ∧ b, a b, a ∨ b and a ⊕ b ∈ I,

(i6)

an ideal of A is also an ideal of L(A),

(i7) Id (A) is partially ordered by set theoretical inclusion, (i8)

if ord (x) = ∞, then there exists a proper ideal I such that x ∈ I.

Chapter I: Łukasiewicz Logic and MV-Algebras

23

The next result can be easily proved. PROPOSITION 2.7.3. Let F be a nonempty subset of A. Then the following conditions are equivalent: (a)

1 ∈ F and for any a, b ∈ A if a ∈ F and a → b ∈ F , then b ∈ F ,

(b)

F satisfies the conditions below (b1) and (b2): (b1) (b2)

(c)

for any a, b ∈ A if a ∈ F and a ≤ b, then b ∈ F , if a ∈ F and b ∈ F , then a b ∈ F ,

the set F ∗ = {a∗ | a ∈ F } is an ideal of A.

DEFINITION 2.7.4. A nonempty subset F of A is a filter if it satisfies one of the equivalent conditions from Proposition 2.7.3. LEMMA 2.7.5. Let A be an MV-algebra and I an ideal of A. Then the MV-subalgebra generated by I in A is hIi = I ∪ I ∗ . Proof. I ∪ I ∗ is obviously closed to the unary operation ∗ and 0 ∈ I. We only have to prove that I ∪ I ∗ is also closed to ⊕. Since I is an ideal and I ∗ is a filter of A, both I and I ∗ are closed to ⊕. If x ∈ I and y ∈ I ∗ , then x ⊕ y ≥ y, so x ⊕ y ∈ I ∗ . Hence I ∪ I ∗ is an MV-subalgebra of A and it is obviously generated by I. DEFINITION 2.7.6. If A and B are two MV-algebras, then a homomorphism is a function f : A → B which satisfies the following conditions: (M1) f (0) = 0, (M2) f (a ⊕ b) = f (a) ⊕ f (b) for any a, b ∈ A, ∗

(M3) f (a∗ ) = f (a) for any a ∈ A. An injective homomorphism is called embedding. We say that the MV-algebra A is embedded in the MV-algebra B if there is an embedding f : A → B. A homomorphism which is also a bijective function is called isomorphism. We say that the MV-algebras A and B are isomorphic if there is an isomorphism f : A → B. We denote A ' B whenever A and B are isomorphic MV-algebras. FACT 2.7.7. Let f : A → B be an MV-algebra homomorphism. One can immediately prove that: (a)

f (1) = 1,

(b)

f (a b) = f (a) f (b),

(c) f (a ∨ b) = f (a) ∨ f (b), (d)

f (a ∧ b) = f (a) ∧ f (b),

(e) f (a → b) = f (a) → f (b), for any a, b ∈ A. Thus, f is also a lattices homomorphism from L(A) to L(B). In particular, f is an increasing function.

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EXAMPLE 2.7.8. Let ∗ [0, 1] be the MV-algebra from Example 2.5.3 and τ ∈ ∗ [0, 1] an infinitesimal. One can easily prove that the set {nτ | n ∈ N} ∪ {[1] − nτ | n ∈ N} is an MV-subalgebra of ∗ [0, 1] which is isomorphic to Chang’s MV-algebra C from Example 2.4.5. FACT 2.7.9. If f : A → B is an MV-algebra homomorphism then, by Remark 2.7.7 (e), f is also a Wajsberg algebra homomorphism between WA and WB . Moreover, a Wajsberg algebras homomorphism is also an MV-algebra homomorphism between the corresponding MV-algebras structures. These facts, together with Corollary 2.3.11, assert that there is a categorical equivalence between the category of MV-algebras and the category of Wajsberg algebras. If f : A → B is an MV-algebra homomorphism, then the kernel of f is ker (f ) = f −1 (0) = {a ∈ A | f (a) = 0}. PROPOSITION 2.7.10. If f : A → B is an MV-algebra homomorphism, then the following assertions hold: (a)

ker (f ) is an ideal of A,

(b)

f is injective iff ker (f ) = {0},

(c)

if J ⊆ B is an ideal, then f −1 (J) is an ideal of A and ker (f ) ⊆ f −1 (J),

(d)

if f is surjective and I ⊆ A is an ideal such that ker (f ) ⊆ I, then f (I) is an ideal of B.

Proof. (a) Since f is a homomorphism we have f (0) = 0, so 0 ∈ ker (f ). If a, b ∈ A such that a ≤ b and b ∈ ker (f ), then f (a) ≤ f (b) and f (b) = 0. We get f (a) = 0, so a ∈ ker (f ). If a, b ∈ ker (f ), then f (a ⊕ b) = f (a) ⊕ f (b) = 0 ⊕ 0 = 0, so a ⊕ b ∈ ker (f ). Hence, ker (f ) is an ideal. (b) We suppose that f is injective and let a ∈ ker(f ). Then f (a) = f (0) = 0, so a = 0. Conversely, let ker (f ) = {0} and a, b ∈ A such that f (a) = f (b). It follows that f (d(a, b)) = d(f (a), f (b)) = 0. Since ker (f ) = {0} we get d(a, b) = 0, so a = b. Thus, f is an injective homomorphism. (c) If J is an ideal of B, then 0 ∈ J. Hence, f (a) = 0 ∈ J for any a ∈ ker (f ), so ker (f ) ⊆ J. Let a1 , a2 ∈ A such that a1 ≤ a2 and a2 ∈ f −1 (J). It follows that f (a1 ) ≤ f (a2 ) and f (a2 ) ∈ f (f −1 (J)) ⊆ J, so f (a2 ) ∈ J. Thus, a2 ∈ f −1 (J). Similarly, if a1 , a2 ∈ f −1 (J), then f (a1 ), f (a2 ) ∈ f (f −1 (J)) ⊆ J, so f (a1 ⊕ a2 ) = f (a1 ) ⊕ f (a2 ) ∈ J and a1 ⊕ a2 ∈ f −1 (J). We proved that f −1 (J) is an ideal of A. (d) Since 0 ∈ I, we get 0 = f (0) ∈ f (I), so f (I) is not empty. Let b1 ≤ b2 ∈ B and b2 ∈ f (I). It follows that there is a2 ∈ I such that f (a2 ) = b2 . Because f is surjective, we also find an element a1 ∈ A such that f (a1 ) = b1 . Thus, f (a1 ) ≤ f (a2 ) which implies that f (a1 a∗2 ) = f (a1 ) f (a2 )∗ = 0. Hence a1 a∗2 ∈ Ker(f ) ⊆ I and a2 ∈ I, so a2 ⊕ a1 a∗2 = a2 ∨ a1 is in I. It follows that a1 ∈ I, which means that b1 = f (a1 ) ∈ f (I). Moreover, if f (a1 ), f (a2 ) are in f (I) for some a1 , a2 ∈ I, then a1 ⊕ a2 ∈ I, so f (a1 ) ⊕ f (a2 ) = f (a1 ⊕ a2 ) is also in f (I). We proved that f (I) is an ideal of B.

Chapter I: Łukasiewicz Logic and MV-Algebras

25

COROLLARY 2.7.11. If f : A → B is a surjective MV-algebra homomorphism, then there is a bijective correspondence between {I | I ∈ Id (A), ker (f ) ⊆ I} and Id (B). Proof. By Proposition 2.7.10 (c) and (d), the desired correspondence is given by J 7→ f −1 (J) and I 7→ f (I), for any J ∈ Id (B) and I ∈ Id (A) such that ker (f ) ⊆ I. We have to prove that f (f −1 (J)) = J and f −1 (f (I)) = I. The first relation is satisfied because f is surjective. Since I ⊆ f −1 (f (I)) always holds, we only have to prove the converse inclusion. If a ∈ f −1 (f (I)), then f (a) ∈ f (I), so there is x ∈ I such that f (a) = f (x). It follows that f (a x∗ ) = f (a) f (x)∗ = 0, so a x∗ ∈ ker (f ) ⊆ I. Hence x ∈ I and a x∗ ∈ I. We get a ∨ x = x ⊕ a x∗ ∈ I, which implies that a ∈ I. Our proof is now complete. LEMMA 2.7.12. If A is an MV-algebra, f : A → A an MV-algebra homomorphism and a ∈ A such that a = a∗ , then f (a) = a. ∗

Proof. Since f is a homomorphism, we have f (a) = f (a∗ ) = f (a). Thus, a and f (a) ∗ are elements of A such that a = a∗ and f (a) = f (a) . Using Proposition 2.1.4 we infer that f (a) = a. EXAMPLE 2.7.13 (The ideals and the homomorphisms of [0, 1]). Let [0, 1] be the MValgebra from Example 2.4.2 and I ⊆ [0, 1] and ideal. Suppose that there is a ∈ I such n times

that a 6= 0. It follows that there is n ∈ N such that a+ · · · +a ≥ 1, where + denotes the real numbers addition. We get na = 1, so ord (a) < ∞. Since I is an ideal, then na = 1 ∈ I and I = [0, 1]. We conclude that Id ([0, 1]) = {{0}, [0, 1]}. Now, let f : [0, 1] → [0, 1] be an MV-algebra homomorphism. By Proposition 2.7.10 (a), ker (f ) is a proper ideal of [0, 1], so ker (f ) = {0}. Thus, by Proposition 2.7.10 (b), f is injective. Since 1/2∗ = 1 − 1/2 = 1/2 we get f (1/2) = 1/2 by Lemma 2.7.12. Let m > 2 be an even natural number and let k = m/2. We have k(1/m) = 1/2, so kf (1/m) = f (1/2) = 1/2, since f is a homomorphism. Recall that a ⊕ b = min{a + b, 1} for any a, b ∈ [0, 1], i.e. a ⊕ b = a + b if a + b ≤ 1 and a ⊕ b = 1 otherwise. Hence, kf (1/m) = 1/2, so f (1/m) = (1/2)/k = (1/2)/(m/2) = 1/m. If m > 2 is an odd natural number, then 1/m = (1/2m) ⊕ (1/2m), so f (1/m) = f (1/2m) ⊕ f (1/2m) = (1/2m) ⊕ (1/2m) = 1/m. We proved that f (1/m) = 1/m for any m ∈ N such that m 6= 0. Consider n, m ∈ N such that m 6= 0 and n ≤ m. It follows that f (n/m) = f (n(1/m)) = nf (1/m) = n(1/m) = n/m. Thus for any rational number r ∈ [0, 1] we get f (r) = r. Let a ∈ [0, 1] be an arbitrary real number. We know that there are two sequences of rational numbers (rn )n∈N ⊆ [0, 1] and (sn )n∈N ⊆ [0, 1] such that rn ≤ a ≤ sn , rn % a and sn . a. Since f is increasing, we get rn = f (rn ) ≤ f (a) ≤ f (sn ) = sn , so limn→∞ rn ≤ f (a) ≤ limn→∞ sn . Thus, a ≤ f (a) ≤ a and f (a) = a. We proved that f (a) = a for any a ∈ [0, 1]. In conclusion, the only MV-algebra homomorphism f : [0, 1] → [0, 1] is the identity. A similar conclusion can be obtained if we consider the MV-algebra Z ∩ [0, 1] or the MV-algebras Łn with n ≥ 2.

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2.8

The Boolean center Let A be an MV-algebra.

PROPOSITION 2.8.1. If a ∈ A, then the following are equivalent: (a)

a ⊕ a = a,

(b)

a ∧ a∗ = 0,

(c) a ∨ a∗ = 1, (d)

a a = a.

Proof. (a) ⇒ (b) a ∧ a∗ = a∗ (a∗∗ ⊕ a) = a∗ (a ⊕ a) = a∗ a = 0. (b) ⇒ (c) 1 = 0∗ = a ∧ a∗ ∗ = a∗ ∨ a. (c) ⇒ (d) a = a 1 = a (a ∨ a∗ ) = (a a) ∨ (a a∗ ) = (a a) ∨ 0 = a a. (d) ⇒ (a) By hypothesis, a∗ ⊕ a∗ = a∗ . It follows that a = a ⊕ 0 = a ⊕ a a∗ = a ⊕ a (a∗ ⊕ a∗ ) = a ⊕ (a ∧ a∗ ) = (a ⊕ a) ∧ (a ⊕ a∗ ) = (a ⊕ a) ∧ 1 = a ⊕ a. DEFINITION 2.8.2. We denote B(A) = {a ∈ A | a ⊕ a = a}. By Proposition 2.8.1, B(A) is the set of all the complemented elements with respect to the lattice structure of A. We will call B(A) the Boolean center of A. Since, in any bounded distributive lattice, the sublattice of all the complemented elements is a Boolean algebra, it follows that hB(A), ∨, ∧, ∗ , 0, 1i is a Boolean algebra. LEMMA 2.8.3. If a ∈ B(A), then a ⊕ b = a ∨ b and a b = a ∧ b for any b ∈ A. Proof. We will prove that a⊕b ≤ a∨b and a∧b ≤ a b, since the converse inequalities hold by Proposition 2.2.5 (a). We have (a ⊕ b) (a ∨ b)∗ = (a ⊕ b) (a∗ ∧ b∗ ) = ((a ⊕ b) a∗ ) ∧ ((a ⊕ b) b∗ ) = b ∧ a∗ ∧ a ∧ b∗ = 0, ∗

(a ∧ b) (a b) = (a ∧ b) (a∗ ⊕ b∗ ) = (a (a∗ ⊕ b∗ )) ∧ (b (a∗ ⊕ b∗ )) = a ∧ b∗ ∧ b ∧ a∗ = 0. The desired inequalities follows by Proposition 2.2.1 (b). FACT 2.8.4. If a ∈ B(A), then in the interval MV-algebra A(0, a) the MV-algebra operations are defined by: x ⊕[0,a] y = x ⊕ y,

x [0,a] y = x y,

and x∗[0,a] = a ∧ x∗ ,

for any x and y ∈ A(0, a). LEMMA 2.8.5. If a ∈ B(A), then the function f : A → A(0, a) defined as f (x) = x∧a is an MV-algebra homomorphism. Proof. We have f (0) = 0 ∧ a = 0. If x, y ∈ A, then f (x∗ ) = x∗ ∧ a = x∗[0,a] and f (x ⊕ y) = (x ⊕ y) ∧ a = (x ⊕ y) ⊕ a = (x ⊕ y) ⊕ a ⊕ a = (x ⊕ a) ⊕ (y ⊕ a) = f (x) ⊕ f (y) = f (x) ⊕[0,a] f (y), so f is an MV-algebra homomorphism.

Chapter I: Łukasiewicz Logic and MV-Algebras

27

PROPOSITION 2.8.6. Let n ∈ N and a1 , . . . , an ∈ B(A) such that ai ∧ aj = 0 for any i 6= j and a1 ∨ · · · ∨ an = 1. Then A is isomorphic with the direct product of the family {A(0, ai ) | i ∈ {1, . . . , n}} via the isomorphism f defined as f (x) = hx ∧ a1 , . . . , x ∧ an i. Proof. We recall that the operations of the direct product are defined on components. The function f is an MV-algebra homomorphism since, by Lemma 2.8.5, the morphism conditions are satisfied on each component. If x, y ∈ A such that f (x) = f (y), then x∧ai = y∧ai for any i ∈ {1, . . . , n}. We get x = x∧1 = x∧(a1 ∨· · ·∨an ) = (x∧a1 )∨ · · ·∨(x∧an ) = (y∧a1 )∨· · ·∨(y∧an ) = y∧(a1 ∨· · ·∨anQ ) = y, so f is injective. In order to prove the surjectivity, we consider hx1 , . . . , xn i ∈ {A(0, ai ) | i ∈ {1, . . . , n}}. Note that xi ∧ ai = xi for any i ∈ {1, . . . , n} and xi ∧ aj ≤ ai ∧ aj = 0, so xi ∧ aj = 0 for any i 6= j. If x = x1 ∨ · · · ∨ xn , then x ∧ ai = (x1 ∧ ai ) ∨ · · · ∨ (xn ∧ an ) = xi . Thus, f (x) = hx1 , . . . , xn i and f is surjective. We have proved that f is an MV-algebra isomorphism. An MV-algebra A is indecomposable if A is nontrivial and A ' A1 × A2 implies either A1 or A2 is trivial. PROPOSITION 2.8.7. An MV-algebra A is indecomposable iff B(A) = {0, 1}. Proof. Let A be an indecomposable MV-algebra and suppose that there is a ∈ B(A) \ {0, 1}. We have a ∨ a∗ = 1 and a ∧ a∗ = 0 so, by Proposition 2.8.6, A is isomorphic to the direct product A(0, a) × A(0, a∗ ). Since a 6= 0 and a∗ 6= 0, neither A(a) or A(a∗ ) are trivial so we get a contradiction with the hypothesis that A is indecomposable. Conversely, let A be an MV-algebra such that B(A) = {0, 1} and suppose that A ' A1 × A2 where A1 and A2 are nontrivial MV-algebras. It follows that the element h0, 1i ∈ A1 × A2 is in the Boolean center. Thus, there is a Boolean element in B(A) \ {0, 1}, which is a contradiction. We proved that A is a indecomposable MV-algebra. EXAMPLE 2.8.8. The MV-algebra [0, 1] and Chang’s MV-algebra C are indecomposable. 2.9

The Riesz decomposition property

Let hA, ⊕, ∗ , 0i be an MV-algebra. We will define a partial binary operation + on A as follows: for any a, b ∈ A, a + b is defined iff a ≤ b∗ and, in this case, a + b := a ⊕ b. One can easily see that a ≤ b∗ iff b ≤ a∗ , so a + b is defined iff b + a is defined and, in this case, a + b = b + a. EXAMPLE 2.9.1. If A is one of the MV-algebras from Example 2.4.2 ([0, 1], Z ∩ [0, 1], or Łn ), then the partial addition + on A is just the real numbers addition. NOTATION 2.9.2. For any a ∈ A we will denote (0 a)+ = 0. For any n ≥ 1, if ((n − 1)a)+ is defined and ((n − 1)a)+ + a is defined, then we will denote (na)+ = ((n − 1)a)+ + a.

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PROPOSITION 2.9.3. For any a, b, c ∈ A the following properties hold: (a)

a + 0 = a,

(b)

a + a∗ = 1,

(c) a ∨ b = a + (a∗ b), (d)

if a + b and (a + b) + c are defined, then b + c and a + (b + c) are defined and (a + b) + c = a + (b + c),

(e)

if a + b = 1, then b = a∗ ,

(f)

if a + b = c, then b = a∗ c.

Proof. (a) Since a ≤ 1 = 0∗ we have a + 0 = a ⊕ 0 = a. (b) Since a ≤ a = (a∗ )∗ we have a + a∗ is defined and a + a∗ = a ⊕ a∗ = 1. (c) Since a ≤ a⊕b∗ = (a∗ b)∗ we know that a+(a∗ b) is defined and a+(a∗ b) = a ⊕ (a∗ b) = a ∨ b. (d) Since (a + b) + c is defined, we have a + b ≤ c∗ which is equivalent to c ≤ a∗ b∗ . It follows that b ≤ a ⊕ b = a + b ≤ c∗ , so b + c is defined. Moreover, b + c = b ⊕ c ≤ b ⊕ (a∗ b∗ ) = b ∨ a∗ . Because a + b is defined, we get b ≤ a∗ . Thus, b + c ≤ a∗ and a + (b + c) is also defined. Finally, we have (a + b) + c = a + (b + c) = a ⊕ b ⊕ c. (e) If a + b is defined and a + b = 1, then b ≤ a∗ and a ⊕ b = 1. We get a∗ ≤ b using Proposition 2.2.1 (a), so b = a∗ . ( f ) If a + b is defined and a + b = c, then b ≤ a∗ and a ⊕ b = c. It follows that b = b ∧ a∗ = a∗ (a ⊕ b) = a∗ c. LEMMA 2.9.4. If a, b ∈ A, then a ≤ b iff there is c ∈ A such that a + c = b. Moreover, if a ≤ b, then c = a∗ b is the unique element of A with the property that a + c = b. Proof. If a + c is defined and a + c = b, then a ≤ a ⊕ c = a + c = b. Conversely, let us suppose that a ≤ b and define c = a∗ b. It follows that a ≤ a ⊕ b∗ = c∗ , so a + c is defined and a + c = a ⊕ c = a ∨ b = b. The uniqueness of c follows by Proposition 2.9.3 (f). PROPOSITION 2.9.5. For any a, b, x, y ∈ A, the following cancellation properties hold: (a)

if a + x = a + y, then x = y,

(b)

if a + x ≤ a + y, then x ≤ y,

(c)

if a ≤ x, b ≤ y and a + b = x + y, then a = x and b = y.

Proof. (a) Since a + x and a + y exist, we get x ≤ a∗ and y ≤ a∗ . Then x = x ∧ a∗ = a∗ (a ⊕ x) = a∗ (a + x) = a∗ (a + y) = a∗ (a ⊕ y) = a∗ ∧ y = y. (b) If a + x ≤ a + y then, by Lemma 2.9.4, there is b ∈ A such that a + x + b = a + y. Using (a), we get x + b = y. Our conclusion follows by Lemma 2.9.4. (c) By Lemma 2.9.4, there are v, w ∈ A such that x = a + v and y = b + w. It follows that a + b = x + y = a + b + v + w and, by (a), we get v + w = 0. Hence v = w = 0, which proves that a = x and b = y.

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29

PROPOSITION 2.9.6. In any MV-algebra A, the following are equivalent: (a)

if c ≤ a + b, then there are a1 , b1 ∈ A such that a1 ≤ a, b1 ≤ b and c = a1 + b1 ,

(b)

if x + y = a + b, then there are z11 , z12 , z21 , z22 ∈ A such that x = z11 + z12

a = z11 + z21

y = z21 + z22

b = z12 + z22 .

Proof. (a) ⇒ (b) If x + y = a + b, then x ≤ a + b. By hypothesis, we infer that there are z11 , z12 ∈ A such that z11 ≤ a, z12 ≤ b and x = z11 + z12 . We define z21 = z11 ∗ a and z22 = z12 ∗ b. By Lemma 2.9.4, z21 is the unique element of A such that z11 + z21 = a and z22 is the unique element of A such that z12 + z22 = b. Since x + y = a + b, it follows that z11 + z12 + y = z11 + z21 + z12 + z22 . Using Proposition 2.9.5 (b) we get y = z21 + z22 and the desired conclusion is proved. (b) ⇒ (a) If c ≤ a + b, then, by Lemma 2.9.4, there is d ∈ A such that c + d = a + b. By hypothesis there are z11 , z12 , z21 , z22 ∈ A such that c = z11 + z12 , a = z11 + z21 and b = z12 + z22 . If we consider a1 = z11 and b1 = z12 , then the desired result is straightforward. DEFINITION 2.9.7 (Riesz decomposition property). We say that an MV-algebra A has the Riesz decomposition property if one of the equivalent conditions from Proposition 2.9.6 is satisfied. PROPOSITION 2.9.8. Any MV-algebra A has the Riesz decomposition property. Proof. We will prove that any MV-algebra A satisfies the condition (a) from Proposition 2.9.6. Let c ≤ a + b and consider a1 = a ∧ (c b∗ ). Then a1 ≤ a. Moreover, a1 c∗ = (a c∗ ) ∧ (c b∗ c∗ ) = (a c∗ ) ∧ 0 = 0, so a1 ≤ c. By Lemma 2.9.4, we infer that a2 = a1 ∗ c is the unique element such that c = a1 + a2 . To complete our proof, we must show that a2 ≤ b. Note that c ≤ a + b implies c a∗ b∗ = 0. Thus, we get a2 b∗ = c b∗ a1 ∗ = c b∗ (a∗ ∨(c∗ ⊕b)) = (c b∗ a∗ )∨((c b∗ ) (c∗ ⊕b)) = 0 ∨ ((c b∗ ) (c b∗ )∗ ) = 0 ∨ 0 = 0, so a2 ≤ b.

3

Ideals in MV-algebras

Throughout this section, A will be an MV-algebra. We will investigate the set Id (A), of all the ideals of A, as well as Spec(A) (the set of all prime ideals of A) and Max (A) (the set of all maximal ideals of A). It is proved that Id (A) is a Brouwerian lattice and that the ideals of A are in bijective correspondence with the congruences of A. Some classical results (the first and the second isomorphism theorem, the Chinese remainder theorem, a subdirect representation theorem, the prime ideal theorem) are proved in the context of MV-algebras. Section 3.3 is concerned with the analysis of Spec(A), while Section 3.4 deals with Max (A). We also introduce the primary ideals, i.e. those ideals that can be embedded in an unique maximal ideal, and we prove that any prime ideal is primary. The radical of an MV-algebra, defined as usual as the intersection of all the maximal ideals, is investigated in Section 3.5, as well as the finite and infinite elements of an MV-algebra. In Section 3.6 we define in classical manner the spectral topology on Spec(A) and Max (A).

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3.1

The lattice of the ideals of A

We recall we have denoted by Id (A) the set of all the ideals of A. In the following we will describe the ideal generated by a given set of elements, we will further analyze the principal ideals of an MV-algebra A and we will prove that Id (A) is a Brouwerian lattice. LEMMA 3.1.1. If {Ik | k ∈ K} is a family of ideals from A, then the intersection T {Ik | k ∈ K} is also an ideal of A. Hence, Id (A) is closed under arbitrary intersections. T Proof. We denote I = {Ik | k ∈ K}. Obviously, I is not empty because 0 ∈ I. If a ≤ b and b ∈ I, then b ∈ Ik for any k ∈ K. We get a ∈ Ik for any k ∈ K and, thus, a ∈ I. Similarly, if a, b ∈ I, then a, b ∈ Ik for any k ∈ K, so a ⊕ b ∈ Ik for any k ∈ K. It follows a ⊕ b ∈ I. We proved that I is an ideal. DEFINITION 3.1.2. Let S be a subset of A. We will denote by (S ] the ideal generated by S, i.e. the smallest ideal that includes S. If a ∈ A, then the ideal generated by {a} will be simply denoted (a]. An ideal I is called principal if there is a ∈ A such that I = (a]. For any ideal I of A we define W (I) = {a ∈ I | I = (a]}. Obviously, an ideal I is principal iff W (I) 6= ∅. FACT 3.1.3. By Lemma 3.1.1, the ideal (S ] exists for any S ⊆ A. Moreover, (∅] = {0}. PROPOSITION 3.1.4. If S is a nonempty set of A, then (S ] = {a ∈ A | a ≤ x1 ⊕ · · · ⊕ xn for some n ∈ N and x1 , . . . , xn ∈ S}. Proof. If I = {a ∈ A | a ≤ x1 ⊕ · · · ⊕ xn for some n ∈ N and x1 , . . . , xn ∈ S}, then we will prove that I is the smallest ideal containing S. Note that I is not empty because S ⊆ I. Let a ≤ b and b ∈ I, so there are n ∈ N and x1 , . . . , xn ∈ S such that a ≤ b ≤ x1 ⊕ · · · ⊕ xn . It follows that a ∈ I. If a, b ∈ I, then a ≤ x1 ⊕ · · · ⊕ xn and b ≤ y1 ⊕ · · · ⊕ ym for some x1 , . . . , xn , y1 , . . . , ym ∈ S. We get a ⊕ b ≤ x1 ⊕ · · · ⊕ xn ⊕ y1 ⊕ · · · ⊕ ym , so a ⊕ b ∈ I. Thus I is an ideal containing S. Let J be another ideal of A that contains S and let a be an arbitrary element from I. Hence a ≤ x1 ⊕ · · · ⊕ xn and x1 , . . . , xn ∈ S ⊆ J. Because J is an ideal, it follows that x1 ⊕ · · · ⊕ xn ∈ J, so a ∈ J and I ⊆ J. We proved that I = (S ]. COROLLARY 3.1.5. The following assertions hold: (a)

if a ∈ A, then (a] = {x ∈ A | x ≤ na for some n ∈ N},

(b)

if a ∈ A, then (a] is proper iff ord (a) = ∞,

(c) W (A) = {a ∈ A | ord (a) < ∞}, (d)

if a ∈ A, I ∈ Id (A) and a ∈ / I, then (I ∪ {a}] = {x ∈ A | x ≤ b ⊕ na for some n ∈ N and b ∈ I},

(e)

if I, J ∈ Id (A), then (I ∪ J ] = {x ∈ A | x ≤ a ⊕ b for some a ∈ I and b ∈ J}.

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31

Proof. (a) The proof is straightforward by Proposition 3.1.4. (b) The ideal (a] is proper iff 1 ∈ / (a] iff 1 6= na for any n ∈ N iff ord (a) = ∞. (c) Follows directly from (b). (d) If x ∈ (I ∪{a}] then, by Proposition 3.1.4, there are m, n ∈ N and x1 , . . . , xm ∈ I such that x ≤ x1 ⊕ · · · ⊕ xm ⊕ na. If we denote b = x1 ⊕ · · · ⊕ xm , then b ∈ I since I is an ideal. The desired result is now obvious. (e) Follows by Proposition 3.1.4 and the fact that any ideal is closed under finite sums. LEMMA 3.1.6. If I is an ideal of A, then W (I) = {a ∈ I | for any b ∈ I there is n ∈ N such that b ≤ na}. Proof. Let a ∈ W (I) and b ∈ I. It follows that b ∈ I = (a] so, by Corollary 3.1.5 (a), there is n ∈ N such that b ≤ na. Conversely, suppose a ∈ I with the property that for any b ∈ I there is n ∈ N such that b ≤ na. Since a ∈ I, we get (a] ⊆ I. If b ∈ I we know that b ≤ na for some natural number n, so b ∈ (a]. Thus I ⊆ (a], so I = (a]. FACT 3.1.7. If we denote I ∨ J := (I ∪ J ] and I ∧ J := I ∩ J for any two ideals I and J, then hId (A), ∨, ∧i is a bounded lattice in which the first element is {0} and the last element is A. We recall that a lattice hL, ∨, ∧i is called Brouwerian if _ _ x ∧ {yk | k ∈ K} = {x ∧ yk | k ∈ K} for any x ∈ L, W whenever {yk | k ∈ K} exists in L. Obviously, aWBrouwerian lattice isWdistributive. An element x W of L is called compact if, whenever S exists and x ≤ S for S ⊆ L, then x ≤ T for some finite T ⊆ S. The lattice L is compactly generated if every element of L is supremum of compact elements. The lattice L is algebraic if it is complete and compactly generated. PROPOSITION 3.1.8. hId (A), ∨, ∧i is a complete Brouwerian algebraic lattice. Proof. If {Ik | k ∈ K} is a family of ideals from A, then the infimum and the supremum of the family are ^ _ [ {Ik | k ∈ K} = ∩{Ik | k ∈ K} and {Ik | k ∈ K} = ( {Ik | k ∈ K}]. Thus, Id (A) is a complete lattice. Obviously, the compact elements of Id (A) are the finiteWgenerated ideals, i.e. the ideals generated by finite sets. For any ideal I we have I = {(a] | a ∈ I}, so the latticeS Id (A) is algebraic. S In order to prove that Id (A) is Brouwerian weS must show that J ∩( {Ik | k ∈ K}] = ( {J ∩Ik | kS∈ K}]. In fact, we prove that J ∩( {Ik | k ∈ K}] is the S smallest ideal which contains S {J ∩Ik | k ∈ K}. Let U be an ideal of A containing {J ∩ I | k ∈ K} = J ∩ {Ik | k ∈ K}. Thus, k S S J ⊆ SU and {Ik | k ∈ K} ⊆ U , so J ⊆ U and ( {Ik | k ∈ K}] ⊆ u, so J ∩ ( {Ik | k ∈ K}] ⊆ U . Our proof is finished.

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3.2

Congruences and quotient MV-algebras

In the general study of the algebraic structures, the notion of congruence is fundamental. The congruences are in strong connection with the construction of the quotient structures and many representation theorems are based on the characterization of some congruence classes. Using the distance function of an MV-algebra, we prove that there is a bijective correspondence between the lattice of all the congruences defined on the MV-algebra A and the lattice of all the ideals of A. In this section we also prove some classical results of universal algebra, like the first and the second isomorphism theorem, the Chinese remainder theorem, a subdirect representation theorem. Due to the fact that a congruence uniquely determines an ideal (and vice-versa), all this results are expressed using ideals instead of congruences. DEFINITION 3.2.1. An equivalence relation ∼ on A is a congruence if the following properties are satisfied: (C1)

if a ∼ b, then a∗ ∼ b∗ ,

(C2)

if a ∼ b and x ∼ y, then (a ⊕ x) ∼ (b ⊕ y),

for any a, b, x, y ∈ A. We will denote by Con(A) the set of all the congruence relations on A. Obviously, Con(A) is partially ordered by set theoretical inclusion. LEMMA 3.2.2. For any congruence ∼ on A and a, b ∈ A we have: (a)

if a ∼ b and x ∼ y, then a x ∼ b y, a ∨ x ∼ b ∨ y and a ∧ x ∼ b ∧ y,

(b)

if a ⊕ b ∼ 0, then a ∼ 0,

(c) a ∼ b iff d(a, b) ∼ 0. ∗

∗

Proof. (a) a x = (a∗ ⊕ x∗ ) ∼ (b∗ ⊕ y ∗ ) = b y. The desired relations for ∨ and ∧ follows similarly. (b) If a ⊕ b ∼ 0, then a∗ ⊕ a ⊕ b ∼ a∗ , so 1 ∼ a∗ . It follows that a ∼ 0. (c) If a ∼ b, then a b∗ ∼ 0 and b a∗ ∼ 0, so d(a, b) ∼ 0. Conversely, if d(a, b) ∼ 0, by (b), we get a b∗ ∼ 0 and a∗ b ∼ 0. Thus, a ∨ b = b ⊕ a b∗ ∼ b and a ∨ b ∼ a ⊕ b a∗ ∼ a. By transitivity, we infer that a ∼ b. LEMMA 3.2.3. If I is an ideal, then the relation ∼I defined by a ∼I b

iff d(a, b) ∈ I

is a congruence on A. Proof. Firstly we prove that ∼I is an equivalence on A. The relation ∼I is obviously symmetric. The reflexivity follows by the fact that d(a, a) = 0 ∈ I for any a ∈ A. In order to prove the transitivity, we suppose that a ∼I b and b ∼I c, i.e. d(a, b) and d(b, c) are in I. Thus, d(a, b) ⊕ d(b, c) is in I. By Proposition 2.6.1 (d7), we have d(a, c) ≤ d(a, b) ⊕ d(b, c), so d(a, c) ∈ I and a ∼I c. Now we have to prove the congruence properties. If a ∼I b, then d(a, b) = d(a∗ , b∗ ) ∈ I, so a∗ ∼I b∗ . Suppose a ∼I b and x ∼I y, i.e. d(a, b) and d(x, y) are in I. By Proposition 2.6.1 (d8), d(a ⊕ x, b ⊕ y) ≤ d(a, b) ⊕ d(x, y) so d(a ⊕ x, b ⊕ y) ∈ I. Hence, a ⊕ x ∼I b ⊕ y and we have proved that ∼I is a congruence relation on A.

Chapter I: Łukasiewicz Logic and MV-Algebras

33

LEMMA 3.2.4. If ∼ is a congruence on A, then the set I∼ = {a ∈ A | a ∼ 0} is an ideal. Proof. Because ∼ is reflexive we get 0 ∈ I, so I is nonempty. If a ≤ b and b ∈ I, then a = a ∧ b ∼ a ∧ 0 = 0, so a ∈ I. If a and b are in I, then a ⊕ b ∼ 0 ⊕ 0 = 0, so a ⊕ b ∈ I. Hence I is an ideal. PROPOSITION 3.2.5. The partially ordered sets Id (A) and Con(A) are isomorphic via isomorphism θ defined as θ(I) = ∼I . Proof. Let I and J be two ideals such that θI = θJ. If a ∈ A we get a = d(a, 0) ∈ I iff a ∼I 0 iff a ∼J 0 iff a = d(a, 0) ∈ J , so I = J. Thus, θ is injective. The map θ is also surjective since, for any ∼ ∈ Con(A), we have θ(I∼ ) = ∼. Indeed, for any a and b in A, we have aθ(I∼ )b iff d(a, b) ∈ I∼ iff d(a, b) ∼ 0 iff a ∼ b. We finish our proof showing that I ⊆ J iff ∼I ⊆ ∼J for any two ideals I and J. If I ⊆ J and a ∼I b, then d(a, b) ∈ I ⊆ J, so a ∼J b. Conversely, if ∼I ⊆ ∼J and a ∈ I, then a = d(a, 0) ∈ I, so a ∼I 0. It follows that a ∼J 0. Thus a ∈ J and our proof is now complete. If I is an ideal of A and a ∈ A we will denote by [a]I the congruence class of a with respect to ∼I , i.e. [a]I = {b ∈ A | a ∼I b}. One can easily see that a ∈ I iff [a]I = [0]I . We will denote by A/I = {[a]I | a ∈ A} the set of all the congruence classes determined by ∼I . Since ∼I is a congruence relation, the MV-algebra operations on A/I given by [a]I ⊕ [b]I := [a ⊕ b]I

and ([a]I )∗ := [a∗ ]I ,

are well defined. Hence, hA/I, ⊕, ∗ , [0]I i is an MV-algebra which is called the quotient of A by I. The function πI : A → A/I defined by πI (a) = [a]I for any a ∈ A is a surjective homomorphism, which is called the canonical projection from A to A/I. One can easily prove that Ker(πI ) = I. LEMMA 3.2.6. If I is an ideal of A and a, b ∈ A, then the following are equivalent: (a)

[a]I = [b]I ,

(b)

a = (b ⊕ x) y ∗ for some x, y ∈ I.

Proof. (a) ⇒ (b) We denote x = a b∗ and y = b a∗ . By hypothesis, x and y are in I. Note that b ⊕ x = a ∨ b = a ⊕ y, so (b ⊕ x) y ∗ = a ∧ y ∗ . Since a ≤ a ⊕ b∗ = y ∗ , the desired equality follows. ∗ (b) ⇒ (a) [a]I = ([b]I ⊕ [x]I ) [y]I = ([b]I ⊕ [0]I ) [1]I = [b]I . PROPOSITION 3.2.7. If I is an ideal of A, then: (a)

πI (J) is an ideal of A/I, where J is an ideal of A containing I,

(b)

the correspondence J 7→ πI (J) is a bijection between the set of the ideals of A containing I and the set of the ideals of A/I.

Proof. (a) Follows by Proposition 2.7.10 (d). (b) Straightforward by Corollary 2.7.11.

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The well-known isomorphism theorems have corresponding versions for MValgebras. We only enunciate the first and the second isomorphism theorem, since their proof follows directly from the classical ones. THEOREM 3.2.8 (The first isomorphism theorem). If A and B are two MV-algebras and h : A → B is a homomorphism, then A/Ker(h) and h(A) are isomorphic MValgebras. THEOREM 3.2.9 (The second isomorphism theorem). If A is an MV-algebra and I, J are two ideals such that I ⊆ J, then (A/I)/πI (J) and A/J are isomorphic MValgebras. THEOREM 3.2.10 (The Chinese remainder theorem). Let n ≥ 1 and let I1 , . . . , In be ideals of an MV-algebra A such that Ii ∨ Ij = A for any i, j ∈ {1, . . . , n} such that i 6= j. Then for every x1 , . . . , xn ∈ A there is x ∈ A such that [x]Ii = [xi ]Ii for any i ∈ {1, . . . , n}. Proof. If n = 1 the desired result obviously follows, since any two elements are equivalent with respect to I1 = A. We prove the theorem for an arbitrary n ≥ 2. Let i, j ∈ {1, . . . , n} such that i 6= j. Since 1 ∈ A = Ii ∨ Ij , by Remark 3.1.7 and Corollary 3.1.5 (e), there are aij ∈ Ii and aji ∈ Ij such that aij ⊕ aji = 1. Thus, [aij ]Ii = [0]Ii and [aji ]Ii = [1]Ii for any i, j ∈ {1, . . . , n} such that i 6= j. Let x1 , . . . , xn ∈ A and denote yi = xi a1i · · · a(i−1)i a(i+1)i · · · ani for any i ∈ {1, . . . , n}. Let i, j ∈ {1, . . . , n} such that i 6= j. Note that [yi ]Ii = [xi ]Ii [a1i ]Ii · · · [a(i−1)i ]Ii [a(i+1)i ]Ii · · · [ani ]Ii = [xi ]Ii [1]Ii · · · [1]Ii [1]Ii · · · [1]Ii = [xi ]Ii , [yi ]Ij = [xi ]Ij [a1i ]Ij · · · [a(i−1)i ]Ij [a(i+1)i ]Ij · · · [ani ]Ij = [xi ]Ij [a1i ]Ij · · · [aji ]Ij · · · [ani ]Ij = [xi ]Ij [a1i ]Ij · · · [0]Ij · · · [ani ]Ij = [0]Ij , for any j 6= i. Our thesis follows by taking x = y1 ⊕ · · · ⊕ yn . We have [x]Ii = [yi ]Ii = [xi ]Ii for any i ∈ {1, . . . , n}. The following result is a version of a well known theorem due to Birkhoff. PROPOSITIONT3.2.11. Let A be an MV-algebra and {Ik | k ∈ K} a family of ideals of A such that {Ik | k ∈ K} = {0}. Then A is a subdirect product of the family {A/Ik | k ∈ K}.

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35

Proof. It is straightforward to prove that the function Y f: A→ {A/Ik | k ∈ K} defined as f (a) = ([a]Ik )k∈K is obviously a homomorphism of MV-algebras. If f (a) = f (b), then f (d(a, b)) = d(f (a), f (b)) = 0, so d(a, b) ∈ Ik for any k ∈ K. By hypothesis, d(a, b) = 0 so a = b. Thus f is an injective homomorphism. For k ∈ K we denote by πk the corresponding projection. We have to prove that πk ◦ f is a surjective homomorphism. This is obvious since (πk ◦ f )(a) = πk (f (a)) = [a]Ik for every a ∈ A. We conclude that A is a subdirect product of the family {A/Ik | k ∈ K}. COROLLARY 3.2.12. If A is an MV-algebra and I1 , . . . , In are ideals of A such that I1 ∩ . . . ∩ In = {0} and Ii ∨ Ij = A for any i 6= j, then A is isomorphic to the direct product (A/I1 ) × · · · × (A/In ). Proof. We define f : A → (A/I1 ) × · · · × (A/In ) by f (a) = h[a]Ii ii∈{1,...,n} . Using Proposition 3.2.11, f is an injective homomorphism. We only have to prove that f is surjective. For any element x = h[xi ]Ii ii∈{1,...,n} ∈ I1 × · · · × In , by Theorem 3.2.10, there is an element x ∈ A such that [x]Ii = [xi ]Ii for any i ∈ {1, . . . , n}. Thus f (x) = x, so f is surjective. We proved that f is an isomorphism. FACT 3.2.13. Let A be an MV-algebra and F ⊆ A a filter in A. We define a ∼F b iff (a → b) ∧ (b → a) ∈ F, for any a, b ∈ F . One can easily see that a ∼F b iff a → b ∈ F and b → a ∈ F . By Proposition 2.7.3 (c), I = F ∗ = {a∗ | a ∈ F } is an ideal of A. Note that ((a → b) ∧ (b → a))∗ = d(a∗ , b∗ ). It follows that a ∼F b iff a ∼I b. Thus, the congruence relations corresponding to F and I coincide and, consequently, the quotient MV-algebras A/F and A/I coincide. 3.3

Prime ideals

In the structures belonging to the algebra of logic (i.e., algebraic structures that correspond to some logical system), the prime ideals are involved at least in three important matters, in algebra, topology and logic. They are extensively used for proving the algebraic representation theorems, as well as the topological duality results. The duals of the prime ideals (i.e. the prime filters) models the deduction in the corresponding logical system and they are frequently used in the algebraic proofs of the completeness theorems. We further investigate the prime ideals of an MV-algebra and we will prove two important results: the prime extension property and the prime ideal theorem. Finally, we get a subdirect representation theorem for MV-algebras in terms of prime ideals. PROPOSITION 3.3.1. The following properties are equivalent for any ideal P of A: (a)

for any a, b ∈ A, a b∗ ∈ P or a∗ b ∈ P ,

(b)

for any a, b ∈ A, if a ∧ b ∈ P , then a ∈ P or b ∈ P ,

(c)

for any I, J ∈ Id (A), if I ∩ J ⊆ P , then I ⊆ P or J ⊆ P .

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Proof. Let a, b ∈ A. (a) ⇒ (b) Suppose a ∧ b ∈ P and a b∗ ∈ P . It follows that c = (a ∧ b) ⊕ (a b∗ ) = (a ⊕ a b∗ ) ∧ (b ∨ a) ∈ P . Since a ≤ c, we get a ∈ P . Similarly, if a∗ b ∈ P we infer that b ∈ P . (b) ⇒ (c) Let I and J be two ideals of A such that I ∩ J ⊆ P . If we suppose that I 6⊆ P and J 6⊆ P , then there are a ∈ I \ P and b ∈ J \ P . We get a ∧ b ∈ I ∩ J ⊆ P and, by hypothesis, a ∈ P or b ∈ P which is a contradiction. Thus, I ⊆ P or J ⊆ P . (c) ⇒ (a) Let a and b be two arbitrary elements of A. If we consider I = (a∗ b] and J = (b∗ a], then I ∩ J = {0} by Propositions 2.2.11 (a) and 2.2.12 (a). It follows that I ∩ J ⊆ P , so I ⊆ P or J ⊆ P . Hence, a∗ b ∈ P or b∗ a ∈ P . DEFINITION 3.3.2. An ideal of A is prime if it is proper and it satisfies one of the equivalent conditions from Proposition 3.3.1. We will denote by Spec(A) the set of all the prime ideals of A. FACT 3.3.3. If I and P are ideals of A such that I ⊆ P , then one can easily prove that P ∈ Spec(A) iff πI (P ) ∈ Spec(A/I). Thus, there is a bijective correspondence between the prime ideals of A containing I and the prime ideals of A/I. PROPOSITION 3.3.4 (Prime extension property). If P and I are proper ideals of A such that P ⊆ I and P is prime, then I is also prime. Proof. Let a, b ∈ A. Since P is prime it follows that a b∗ ∈ P or a∗ b ∈ P . Because P ⊆ I we get a b∗ ∈ I or a∗ b ∈ I, so I is a prime ideal of A. LEMMA 3.3.5. If {Pt | t ∈ T } ⊆ Spec(A) is totally ordered by inclusion, then T {Pt | t ∈ T } is also a prime ideal. T Proof. By Lemma 3.1.1, P := {Pt | t ∈ T } is an ideal of A. Let a, b ∈ A such that a ∧ b ∈ P and suppose that a ∈ / P and b ∈ / P . Hence there are t1 and t2 ∈ T such that a ∈ / Pt1 and b ∈ / Pt2 . Since Pt1 and Pt2 are prime ideals, it follows that b ∈ Pt1 and a ∈ Pt2 . The family Pt | t ∈ T is totally ordered so Pt1 ⊆ Pt2 or Pt2 ⊆ Pt1 . If Pt1 ⊆ Pt2 , then b ∈ / Pt1 , which is a contradiction. Similarly, we get a contradiction if Pt2 ⊆ Pt1 . It follows that a ∈ P or b ∈ P , so P is a prime ideal of A. PROPOSITION 3.3.6. Let I be a prime ideal of A. Then the set I = {J | I ⊆ J and J is a proper ideal of A} is linearly ordered with respect to set-theoretical inclusion. Proof. We consider J, K ∈ I and we suppose that J 6⊆ K and K 6⊆ J. Thus, there are a ∈ J \ K and b ∈ K \ J. Since I is prime, we get a b∗ ∈ I ⊆ K or a∗ b ∈ I ⊆ J. It follows that a ∨ b = b ⊕ a b∗ ∈ K or a ∨ b = a ⊕ b a∗ ∈ J, so a ∈ K or b ∈ J which is a contradiction. Thus, J ⊆ K or K ⊆ J and I is linearly ordered.

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COROLLARY 3.3.7. Spec(A), partially ordered by set inclusion, is a root system, i.e. a partially ordered set in which the upper bounds of any element form a chain. THEOREM 3.3.8 (Prime ideal theorem). Let I be a proper ideal of A and let S be a nonempty ∧-closed subset of A (i.e. if a, b ∈ S, then a ∧ b ∈ S) such that I ∩ S = ∅. Then there exists a prime ideal P such that I ⊆ P and P ∩ S = ∅. Proof. We define J = {J | J is a proper ideal, I ⊆ J and S ∩ J = ∅}. A routine application of Zorn’s Lemma shows that J has a maximal element P . We will prove that P is prime. Let a, b ∈ A and suppose a b∗ ∈ / P and a∗ b ∈ / P . Thus, the ideals ∗ ∗ J = (P ∪ {a b }] and K = (P ∪ {a b}] are not in J , so there are c ∈ J ∩ S and d ∈ K ∩ S. If we denote u = a b∗ and v = a∗ b, then c ≤ p1 ⊕ n1 u and d ≤ p2 ⊕n2 v for some p1 , p2 ∈ P and n1 , n2 ∈ N. Note that nu∧nv = 0 for any n ∈ N. Let p = p1 ∨ p2 ∈ P and n = max{n1 , n2 }. It follows c ∧ d ≤ (p ⊕ nu) ∧ (p ⊕ nv) = p ⊕ (nu ∧ nv) = p ∈ P , so c ∧ d ∈ P . Since S is ∧-closed we also have c ∧ d ∈ S which contradicts the fact that S ∩ P = ∅. Thus a b∗ ∈ P or a∗ b ∈ P and P is the desired prime ideal. COROLLARY 3.3.9. Any proper ideal I of A can be extended to a prime ideal. Proof. Apply Theorem 3.3.8 for I and S = {1}. COROLLARY 3.3.10. If a ∈ A, then ord (a) < ∞ iff a ∈ / P for any P ∈ Spec(A). Proof. Assume that ord (a) < ∞ and there exists P ∈ Spec(A) such that a ∈ P . Then for some n ∈ N, na = 1 ∈ P , so P is not a proper ideal, which is impossible. Conversely, assume that a ∈ / P for any P in Spec(A) and ord (a) = ∞. Hence (a] is a proper ideal and we get a contradiction using Corollary 3.3.9. PROPOSITION 3.3.11. If a 6= 0 there is a prime ideal P such that a ∈ / P. Proof. Apply Theorem 3.3.8 for I = {0} and S = {a}. COROLLARY 3.3.12. In every MV-algebra A the intersection of all the prime ideals of A is {0}. Proof. By Proposition 3.3.11. PROPOSITION 3.3.13. Any MV-algebra A is a subdirect product of the family {A/P | P ∈ Spec(A)}. Proof. By Proposition 3.2.11 and Corollary 3.3.12. 3.4

Maximal ideals

In this section we characterize the maximal ideals of an MV-algebra and we prove that any prime ideal can be embedded in an unique maximal ideal (Proposition 3.4.5). DEFINITION 3.4.1. An ideal M of A is maximal if it is a maximal element in the partially ordered set of all the proper ideals of A. This means that M is proper and, for any proper ideal I, if M ⊆ I, then M = I. We will denote by Max (A) the set of all the maximal ideals of A.

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PROPOSITION 3.4.2. If M is a proper ideal of A, then the following are equivalent: (a)

M is maximal,

(b)

for any a ∈ A if a ∈ / M , then there is n ∈ N such that (a∗ )n ∈ M .

Proof. (a) ⇒ (b) Let a be an element of A such that a ∈ / M . Since M is maximal, the ideal (M ∪ {a}] coincide with A. Hence, there is b ∈ M and n ∈ N such that ∗ 1 = b ⊕ na, so (a∗ )n = (na) ≤ b. We get (a∗ )n ∈ M . (b) ⇒ (a) Suppose that I is an ideal of A such that M ⊆ I and M 6= I. Then there exists an element a ∈ I \ M . By hypothesis, (a∗ )n ∈ M for some n ∈ N. It follows that na ∈ I and (na)∗ = (a∗ )n ∈ M ⊆ I, so I = A. FACT 3.4.3. If I and M are ideals of A such that I ⊆ M , then one can easily prove: M ∈ Max (A) iff πI (M ) ∈ Max (A/I). Thus, there is a bijective correspondence between the maximal ideals of A containing I and the maximal ideals of A/I. LEMMA 3.4.4. Any maximal ideal of an MV-algebra is a prime ideal. Proof. Let M be a maximal ideal of A. Because M is a proper ideal of A, there is a prime ideal P of A such that M ⊆ P . Since P is proper, it follows that M = P . Hence, M is prime. PROPOSITION 3.4.5. Any proper ideal of A can be extended to a maximal ideal. Moreover, for any prime ideal of A there is a unique maximal ideal containing it. Proof. Let I be a proper ideal of A. By Corollary 3.3.9, there is a prime ideal P of A such that I ⊆ P . Let P = {J | P ⊆ S J and J is a proper ideal ofA}. By Proposition 3.3.6, P is linearly ordered, so U = {J | J ∈ P} is a proper ideal and I ⊆ U . We will prove that U is a maximal ideal of A. Let H be a proper ideal of A such that U ⊆ H. It follows that H ∈ P, hence H ⊆ U . We get H = U , so U is a maximal ideal which includes I. Let P be a prime ideal and suppose that there are two maximal ideals M and U such that P ⊆ M and P ⊆ U . It follows that M , U ∈ P which is a linearly ordered set, so M ⊆ U or U ⊆ M . Since both U and M are maximal ideal we infer that U = M . Thus, any prime ideal is contained in a unique maximal ideal. 3.5

The radical

In the following, the radical Rad (A) of the MV-algebra A is defined as the intersection of its maximal ideals. This notion is found in algebra especially in ring theory. Since the MV-algebra operation ⊕ is cancellative on Rad (A), it coincides with the partial operation + from Section 2.9. The nonzero elements of Rad (A) are called infinitesimals, since they are characterized by a property that defines the infinitesimals in nonstandard analysis (see Example 2.5.3). DEFINITION 3.5.1. The intersection of the maximal ideals of A is called the radical of A. It will be denoted by Rad (A). It is obvious that Rad (A) is an ideal, since an intersection of ideals is also an ideal.

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LEMMA 3.5.2. For any a, b ∈ Rad (A), the following identities hold: (a)

a b = 0,

(b)

a ⊕ b = a + b,

(c) a ≤ b∗ . Proof. (a) Let a, b ∈ Rad (A) and suppose a b 6= 0. By Proposition 3.3.11 there is a prime ideal P such that a b ∈ / P . Since P is prime it follows that a∗ b∗ ∈ P . By Proposition 3.4.5 there is a maximal ideal M such that P ⊆ M . Thus a ⊕ b = ∗ (a∗ b∗ ) ∈ / M so (M ∪ {a ⊕ b}] = A and 1 = x ⊕ n(a ⊕ b) for some x ∈ M and ∗ n ∈ N. We get (n(a ⊕ b)) ≤ x. If we denote c = n(a ⊕ b), then we infer that c and c∗ are in M which contradicts the fact that M is a proper ideal. (b) By (a), a ≤ b∗ , so a + b is defined and a + b = a ⊕ b. (c) Equivalent with (a) by Proposition 2.2.1. COROLLARY 3.5.3. hRad (A), ⊕, 0i is a naturally ordered cancellative monoid. DEFINITION 3.5.4. An element a of A is called infinitesimal if a 6= 0 and na ≤ a∗ for any n ∈ N. An element x of A is called finite if ord (x) < ∞ and ord (x∗ ) < ∞. FACT 3.5.5. Using Notation 2.9.2, we can equivalently define an infinitesimal as an element a ∈ A such that (na)+ is defined for any n ∈ N. PROPOSITION 3.5.6. For any a ∈ A, a 6= 0, the following are equivalent: (a)

a is infinitesimal,

(b)

a ∈ Rad (A),

(c) (na)2 = 0 for every n ∈ N. Proof. (a) ⇒ (b) Let a be an infinitesimal and suppose a ∈ / Rad (A). Thus, there is a maximal ideal M of A such that a ∈ / M so (M ∪ {a}] = A. We get 1 = na ⊕ b for some ∗ ∗ n ∈ N and b ∈ M . It follows (na) ≤ b ∈ M so (na) ∈ M . By hypothesis na ≤ a∗ , ∗ so a ≤ (na) . We conclude that a ∈ M which is a contradiction. Thus a ∈ Rad (A). (b) ⇒ (c) Since Rad (A) is an ideal it follows that na ∈ Rad (A) for any n ∈ N. The desired result follows by Lemma 3.5.2. (c) ⇒ (a) If n = 0, then na = 0 ≤ a∗ . If n ≥ 1, then (na) a ≤ (na)2 = 0 so (na) a = 0. We get na ≤ a∗ for any n ∈ N. Thus a is an infinitesimal. COROLLARY 3.5.7. If B is an MV-subalgebra of A, then Rad (B) = Rad (A) ∩ B. COROLLARY 3.5.8. Let f : A → B be an MV-algebra homomorphism. Then f (Rad (A)) ⊆ Rad (f (A)). COROLLARY 3.5.9. If B is MV-algebra, then Rad (A × B) = Rad (A) × Rad (B). PROPOSITION 3.5.10. If a is an infinitesimal of A, then the set ∗

C(a) = {na | n ∈ N} ∪ {(na) | n ∈ N} is a subalgebra of A, isomorphic to Chang’s algebra C.

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Proof. By Lemma 3.5.2 we get (na) (ma) = 0, so na ≤ (ma) for any n, m ∈ N. In ∗ order to prove that C(a) is a subalgebra, it is sufficient to show that na ⊕ (ma) ∈ C(a) ∗ for any n, m ∈ N. If n ≤ m, then na ⊕ (ma) = na ⊕ (na ⊕ (m − n)a)∗ = na ⊕ (na)∗ ((m − n)a)∗ = na ∨ ((m − n)a)∗ = ((m − n)a)∗ ∈ C(a). ∗ ∗ If n > m, then na ⊕ (ma) = (n − m)a ⊕ ma ⊕ (ma) = 1 ∈ C(a). Thus, C(a) is a subalgebra of A. An isomorphism f between C(a) and C is defined by: f (a) = c, ∗ f (a∗ ) = 1 − c, f (na) = nc, f ((na) ) = 1 − nc for any n ∈ N. LEMMA 3.5.11. If x ∈ A is a finite element, then the following properties hold for any a, b, c ∈ Rad (A): (a)

a < x < a∗ ,

(b)

x a∗ is a finite element,

(c) a < b∗ (x ⊕ c), (d)

(x ⊕ a) b∗ = x b∗ ⊕ a,

(e) a∗ b∗ = x a∗ ⊕ x∗ b∗ . Proof. Set ord (x) = n. (a) Note that, for every a ∈ Rad (A), n(a ∧ x) = na implies a ∧ x = a, i.e. a < x. Similarly one proves that a < x∗ , so a < x < a∗ . (b) From Lemma 3.2.6 we obtain [x]Rad(A) = [x a∗ ]Rad(A) . Hence n(x a∗ ) ∈ Rad (A)∗ , i.e. ord (x a∗ ) ≤ 2n < ∞. Since ord (x∗ ⊕ a) ≤ ord (x∗ ) < ∞, x a∗ is a finite element of A. (c) Follows by (a), (b) and Lemma 3.2.6. (d) Using (a) and (b) we get (x⊕a) b∗ = (x b∗ ⊕b⊕a) b∗ = (x b∗ ⊕a)∧b∗ = x b∗ ⊕ a. (e) Using (d) we get the following: (x ⊕ a) b∗ = x b∗ ⊕ a (x ⊕ a) ⊕ (x ⊕ a) b∗ a∗ = (x ⊕ a)∗ ⊕ (x b∗ ⊕ a) a∗ (x ⊕ a)∗ ∨ (b∗ a∗ ) = (x∗ a∗ ) ⊕ (x b∗ ∧ a∗ ) b∗ a∗ = (x∗ a∗ ) ⊕ (x b∗ ). ∗



LEMMA 3.5.12. The following are equivalent for any finite element x ∈ A and for any a, b, c, d ∈ Rad (A): (a)

x a∗ ⊕ b = x c∗ ⊕ d,

(b)

b ⊕ c = a ⊕ d.

Proof. (a) ⇒ (b) We have x a∗ ⊕b⊕a⊕c = x c∗ ⊕d⊕a⊕c. By Lemma 3.5.11 (a) follow x ⊕ b ⊕ c = x ⊕ a ⊕ d and b ⊕ c = a ⊕ d. (b) ⇒ (a) Since x⊕b⊕c = x⊕a⊕d, by Lemma 3.5.11 (a), one gets x a∗ ⊕a⊕b⊕c = x c∗ ⊕ c ⊕ a ⊕ d, i.e. x a∗ ⊕ b = x c∗ ⊕ d. LEMMA 3.5.13. If a, b, c, d ∈ Rad (A) and x, y are finite elements such that x ⊕ y is a finite element, then (x ⊕ a) b∗ ⊕ (y ⊕ c) d∗ = (x ⊕ y ⊕ a ⊕ c) (b ⊕ d)∗ .

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41

Proof. Using Lemmas 3.5.11 and 3.5.12 we get: (x ∨ b) ⊕ (y ∨ d) = (x ⊕ y) ∨ (b ⊕ d) x b∗ ⊕ b ⊕ y d∗ ⊕ d = (x ⊕ y) b∗ d∗ ⊕ b ⊕ d (b ⊕ d)∗ (b ⊕ d ⊕ x b∗ ⊕ y d∗ ) = (b ⊕ d)∗ (b ⊕ d ⊕ (x ⊕ y) b∗ d∗ ) (b ⊕ d)∗ ∧ (x b∗ ⊕ y d∗ ) = (b ⊕ d)∗ ∧ (x ⊕ y) b∗ d∗ x b∗ ⊕ y d∗ = (x ⊕ y) b∗ d∗ a ⊕ c ⊕ x b∗ ⊕ y d∗ = a ⊕ c ⊕ (x ⊕ y) b∗ d∗  (x ⊕ a) b∗ ⊕ (y ⊕ c) d∗ = (x ⊕ y ⊕ a ⊕ c) (b ⊕ d)∗ . 3.6

The spectral topology

Let hA, ⊕, ∗ , 0i be an MV-algebra. In this section we define a topology on Spec(A) and we display some properties of Spec(A) and Max (A). For any I ∈ Id (A) we define r(I) := {P ∈ Spec(A) | I 6⊆ P }. PROPOSITION 3.6.1. The following properties hold: (a)

r({0}) = ∅,

(b)

r(A) = Spec(A),

(c) r(I ∧ J) = r(I) ∩ r(J) for any I, J ∈ Id (A), W S (d) r( {Ik | k ∈ K}) = {r(Ik ) | k ∈ K}, for any set {Ik | k ∈ K} ⊆ Id (A), (e) I ⊆ J iff r(I) ⊆ r(J) for any I, J ∈ Id (A), (f)

if X ⊆ A, then {P ∈ Spec(A) | X * P } = r((X]).

Proof. (a) and (b) Obvious. (c) Let I and J be ideals of A. Recall that I ∧ J = I ∩ J. For any prime ideal P , by Proposition 3.3.1, we have I ∩ J ⊆ P iff I ⊆ P or J ⊆ P . It follows that I ∩ J 6⊆ P iff I 6⊆ P and J 6⊆ P , so r(I ∧ J) = r(I) ∩ r(J). W S (d) If {Ik | k ∈ K} is a set of ideals,Wthen {Ik | k ∈ K}S= ( {Ik | k ∈ K}]. For any prime ideal P we have P ∈ r( {Ik | k ∈ K}) iff ( {Ik | k ∈ K}] 6⊆ P iff there S is k ∈ K such that Ik 6⊆ P iff there is k ∈ K such that P ∈ r(Ik ) iff P ∈ {r(Ik ) | k ∈ K}. (e) If I ⊆ J and P ∈ r(I), then I 6⊆ P . It follows that J 6⊆ P , so P ∈ r(J). Conversely, we consider r(I) ⊆ r(J). If J = A, then the desired result is obvious. Thus, we can assume that J is a proper ideal. We suppose that I 6⊆ J. It follows that there is an element a ∈ I \ J. Using Theorem 3.3.8 for J and S = {a}, we infer that there exists a prime ideal P such that J ⊆ P and a ∈ / P . We get J ⊆ P and I 6⊆ P , so P ∈ / r(J) and P ∈ r(I), which is a contradiction. We proved that I ⊆ J. (f) A consequence of the fact that X ⊆ P iff (X] ⊆ P for any P ∈ Spec(A). If we define τ := {r(I) | I ∈ Id (A)}, then hSpecA, τ i becomes a topological space by Proposition 3.6.1 (a), (b), (c) and (d). In the sequel τ will be referred as the spectral topology. PROPOSITION 3.6.2. Let A be an MV-algebra. Then ideals of A are in one to one correspondence with open sets in Spec(A).

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Proof. Let r be the mapping from ideals of A to open sets in Spec(A) defined as follows: r : I ∈ Id (A) → r(I). Thus, trivially, r is surjective. The injectivity follows by Proposition 3.6.1 (e). For any a ∈ A we define r(a) := {P ∈ Spec(A) | a ∈ / P }. LEMMA 3.6.3. The following properties hold: (a)

r(a) = r((a]) for any a ∈ A,

(b)

r(0) = ∅,

(c) r(1) = Spec(A), (d)

r(a ∨ b) = r(a ⊕ b) = r(a) ∪ r(b) for any a, b ∈ A,

(e) r(a ∧ b) = r(a) ∩ r(b) for any a, b ∈ A, S (f) r(I) = {r(a) | a ∈ I} for any I ∈ Id (A). Proof. (a) Straightforward, since for any element a ∈ A and for any ideal P we have a ∈ P iff (a] ⊆ P . (b), (c) Obvious. (d) Let a, b ∈ A. We have P ∈ r(a ∨ b) iff a ∨ b ∈ / P iff a ∈ / P or b ∈ / P iff P ∈ r(a) ∪ r(b). The other equality follows similarly. (e) For any prime ideal P , we have a ∧ b ∈ P iff a ∈ P or b ∈ P . (f) Let P be any prime ideal of A. We have I * P iff there exists a ∈ I such that a∈ / P . Hence P ∈ r(I) iff there exists a ∈ I such that P ∈ r(a). The desired equality is now obvious. PROPOSITION 3.6.4. The family {r(a) | a ∈ A} is a basis for the topology τ on Spec(A). The compact open subsets of Spec(A) are exactly the sets of the form r(a) for some a ∈ A. Proof. By Lemma 3.6.3 (a), r(a) is an open subset of Spec(A) for any a ∈ A. If I ∈ Id (A) then, by Proposition 3.6.3 (f), any open subset of Spec(A) is a union of elements r(a) with a ∈ A. This means that the family {r(a) | a ∈ A} is a basis for the topology τ . Now we will prove that r(a) is a compact S element for any a ∈ A. Let a ∈ AWand {Ik | k ∈ K} ⊆ Id (A) such that r(a) ⊆ W {r(Ik ) | k ∈ K}. Hence r(a) ⊆ r( {Ik | k ∈ K}) and, by Proposition 3.6.1 (e), a ∈ {Ik | k ∈ K}. It follows that there are n ∈ N and ak1 ∈ Ik1 , . . . , akn ∈ Ikn such that a ≤ ak1 ⊕ · · · ⊕ akn . We get r(a) ⊆ r(ak1 ) ∪ · · · ∪ r(akn ) ⊆ r(Ik1 ) ∪ · · · ∪ r(Ikn ), so r(a) is a compact subset ofSSpec(A). Conversely, let r(I) be a compact open subset of Spec(A). Since r(I) = {r(a) | a ∈ I} and r(I) is compact, it follows that there are n ∈ N and a1 , . . . , an ∈ I such that r(I) = r(a1 ) ∪ · · · ∪ r(an ). By Lemma 3.6.3 (d), we infer that r(I) = r(a1 ∨ · · · ∨ an ). Our proof is finished. THEOREM 3.6.5. For any MV-algebra A, the prime ideal space Spec(A) is a compact T0 space with respect to the spectral topology.

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Proof. By Proposition 3.6.4 it follows that Spec(A) = r(1) is compact. Now we have to prove that Spec(A) is a T0 space, which means that for any two distinct prime ideals P 6= Q ∈ Spec(A) at least one of them has an open neighbourhood not containing the other. If P 6= Q, then P \Q 6= ∅ or Q\P 6= ∅. If P \Q 6= ∅, then there exists a ∈ P \Q, so Q ∈ r(a) and P ∈ / r(a). If Q \ P 6= ∅, then there exists b ∈ Q \ P , so P ∈ r(b) and Q∈ / r(b). THEOREM 3.6.6. If A is an MV-algebra, then the following are equivalent: (a)

Spec(A) is connected,

(b)

B(A) = {0, 1}.

Proof. (a) ⇒ (b) Assume that B(A) 6= {0, 1}. Then there exists a ∈ B(A) \ {0, 1}. Since a ∧ a∗ = 0, we have a ∈ P iff a∗ ∈ / P , for any P ∈ Spec(A). It follows that Spec(A) = r(a) ∪ r(a∗ ), r(a) ∩ r(a∗ ) = ∅, r(a) 6= ∅ and r(a∗ ) 6= ∅, which contradicts our hypothesis. Hence B(A) is the two element algebra. (b) ⇒ (a) If Spec(A) is not connected, then Spec(A) = V1 ∪ V2 , where V1 and V2 are non-empty disjoint openSsubsets of Spec(A). Thus there are X, Y ⊆ A such that S V1 = x∈X r(x) and V2 = y∈Y r(y). Since Spec(A) is compact, it follows that there Sn Sm are x1 , . . . , xn ∈ X and y1 , . . . , ym ∈ Y with V1 = i=1 r(xi ) and V2 = j=1 r(yj ). If we set x = x1 ⊕ · · · ⊕ xn and y = y1 ⊕ · · · ⊕ ym then, by Lemma 3.6.3 (d), we get V1 = r(x), V2 = r(y) and Spec(A) = r(x) ∪ r(y). Then, for any P ∈ Spec(A), x ∈ P iff y ∈ / P . Thus x ∧ y ∈ P and x ⊕ y ∈ / P for any P ∈ Spec(A). By Corollaries 3.3.12 and 3.3.10 it follows that x ∧ y = 0 and ord (x ⊕ y) < ∞, so nx ⊕ ny = 1 for some n ∈ N. By Proposition 2.2.12 (a), nx ∧ ny = 0 so, by Proposition 2.2.5 (f), we also get nx ∨ ny = 1. It follows that nx and ny are Boolean elements of A and nx = (ny)∗ . If nx = 0, then x = 0 and V1 = r(0) = ∅, which is impossible. Hence rx = 1, so ry = 0 and V2 = r(0) = ∅ which is also impossible. We proved that Spec(A) is connected. PROPOSITION 3.6.7. Let A be an MV-algebra and S a subset of Spec(A). Then the following are equivalent: (a)

S is closed in Spec(A),

(b)

for every P ∈ Spec(A), P ∈ S iff

T

H∈S

H ⊆ P.

T Proof. (a) ⇒ (b) Assume S is closed, P ∈ Spec(A) and P ⊇ H∈S H. If V is any open set T of Spec(A) containing P , P ∈ r(a) ⊆ V for some a ∈ A. Since a ∈ / P , we get a ∈ / H∈S H. Then, there exists a prime ideal K ∈ S such that a ∈ / K. Hence K ∈ r(a) ∩ S ⊆ (V ∩ S), so every neighborhood of P contains a point of S. It follows that P is a limit point of S. Since S is closed, we have P ∈ S and (b)Tis proved. (b) ⇒ (a) Let P ∈ T Spec(A) be a limit point of S and P ∈ / S. Then H∈S H * P , so there exists an a ∈ ( H∈S H)\P . It follows that P ∈ r(a). But r(a) is a neighborhood of P , so 6 ∅. If Q ∈ (r(a) ∩ S), then Q ∈ S and a ∈ / Q. But we have T r(a) ∩ S = a ∈ ( H∈S H) ⊆ Q, so a ∈ Q which is a contradiction. Thus P ∈ S and S is closed. Let MV be the category of MV-algebras with MV-homomorphisms and Top the category of topological spaces with continuous functions. Then we can prove:

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THEOREM 3.6.8. The mapping Spec : A ∈ MV → Spec(A) ∈ Top defines a contravariant functor. Proof. Indeed, let ϕ : A → B be an MV-algebra homomorphism and define Spec(ϕ) = ϕ−1 . It is easy to check that for any prime ideal Q of B, ϕ−1 (Q) is a prime ideal of A. So Spec(ϕ) maps Spec(B) to Spec(A). Let us show that Spec(ϕ) is continuous. Indeed, for any open set r(I) of Spec(A), with I an ideal of A, we have: (Spec(ϕ)−1 (r(I)) = {Q ∈ Spec(B) | Spec(ϕ)(Q) ∈ r(I)} = {Q ∈ Spec(B) | ϕ−1 (Q) ∈ r(I)} = {Q ∈ Spec(B) | I * ϕ−1 (Q)} = {Q ∈ Spec(B) | ϕ(I) * Q}. By Proposition 3.6.1 (f), {Q ∈ Spec(B) | ϕ(I) * Q} = r((ϕ(I)]), which is an open set of Spec(B). So, Spec(ϕ) is continuous. Moreover, we have: Spec(ψ ◦ ϕ) = Spec(ψ) ◦ Spec(ϕ) with ϕ and ψ MV-algebra homomorphisms. Hence Spec is contravariant. In the sequel, we will characterize the maximal ideal space Max (A) of an MValgebra A. Since Max (A) ⊆ Spec(A), we endow Max (A) with the topology induced by the spectral topology τ on Spec(A). This means that the open sets of Max (A) are s(I) = r(I) ∩ Max (A) = {M ∈ Max (A) | I 6⊆ M }. In consequence, for any a ∈ A and I ∈ Id (A), s(a) = r(a) ∩ Max (A) = {M ∈ Max (A) | a ∈ / M}

and

s(I) =

[

{s(a) | a ∈ I}.

Hence the family {s(a) | a ∈ A} is a basis for the induced topology on Max (A). LEMMA 3.6.9. The following properties hold: (a)

s(0) = ∅,

(b)

s(A) = s(1) = Max (A),

(c) s(I ∩ J) = s(I) ∩ s(J) for any I, J ∈ Id (A), W S (d) s( {Ik | k ∈ K}) = {s(Ik ) | k ∈ K}, for any set {Ik | k ∈ K} ⊆ Id (A), (e) I ⊆ J implies s(I) ⊆ s(J) for any I, J ∈ Id (A), (f)

if X ⊆ A, then {P ∈ Max (A) | X * P } = r((X]),

(g)

s(I) = Max (A) iff I = A for any I ∈ Id (A).

Proof. (a),(b),(c),(d),(f) Follows as in Proposition 3.6.1. (e) Straightforward. (g) One implication follows by (b) and (e). For the other one, suppose that s(I) = Max (A). Then I 6⊆ M for any M ∈ Max (A). By Proposition 3.4.5, I is not a proper ideal so I = A.

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THEOREM 3.6.10. In any MV-algebra A, the maximal ideal space Max (A) is a compact Hausdorff topological space with respect to the topology induced by the spectral topology on Spec(A). S Proof. We firstly prove that Max (A) is compact. If Max (A) = {s(Ik ) | k ∈ K} forWsome family {Ik | k ∈ K} ⊆ Id (A) then, by Lemma 3.6.9 W (d), Max (A) = s( {Ik | kW ∈ K}). by Lemma 3.6.9 (g) it follows that A = {Ik | k ∈ K}. Since 1 ∈ {Ik | k ∈ K}, there are n ∈ N and ak1 ∈ Ik1 , . . . , akn ∈ Ikn such that 1 = ak1 ⊕ · · · ⊕ akn . Using Lemma 3.6.9 (d), we get Max (A) = s(1) ⊆ s(Ik1 ∨ · · · ∨ Ikn ) = s(Ik1 ) ∪ · · · ∪ s(Ikn ). Hence Max (A) is a compact topological space. Let M and N be two distinct maximal ideals. We will prove that there are a, b ∈ A such that M ∈ s(a), N ∈ s(b) and s(a) ∩ s(b) = ∅. Since M 6= N , then M * N and N * M , so there are x ∈ M \ N and y ∈ N \ M . If a = x∗ y, then a ∈ / M since, otherwise x ⊕ a = y ∨ x ∈ M , so y ∈ M which is a contradiction. Similarly, we infer that b = y ∗ x ∈ / N . Hence, we found a, b ∈ A such that M ∈ s(a) and N ∈ s(b). Moreover, s(a) ∩ s(b) = s(a ∧ b) = s(0) = ∅ by Proposition 2.2.11 (a). It follows that Max (A) is Hausdorff. By Proposition 3.4.5, for any prime ideal P ∈ Spec(A) there is a unique maximal ideal MP ∈ Max (A) such that P ⊆ MP . It is straightforward to define an application M : Spec(A) → Max (A) as

M(P ) = MP

for any P ∈ Spec(A). PROPOSITION 3.6.11. M is a continuous function from Spec(A) to Max (A). Proof. Let U be a closed set of Spec(A). We will prove that M(U ) is a closed set of Max (A) and this means that M is a continuous function. If U is closed in Spec(A), then U = Spec(A) \ r(I) for some ideal I of A. Thus, U = {P ∈ Spec(A) | I ⊆ P } and M(U ) = {M(P ) | P ∈ Spec(A), I ⊆ P }. We prove that M(U ) = Max (A) \ s(I). If M ∈ M(U ), then M = MP for some P ∈ Spec(A) such that I ⊆ P . We get I ⊆ P ⊆ M , so M ∈ Max (A) \ s(I). Conversely, if M ∈ Max (A) \ s(I), then I ⊆ M . Since M(M ) = M , we get M ∈ M(U ). We know that Max (A) \ s(I) is a closed set of Max (A), so the desired conclusion is straightforward. PROPOSITION 3.6.12. If A is an MV-algebra, the following are equivalent: (a)

Spec(A) is Hausdorff,

(b)

Spec(A) = Max (A).

Proof. (a) ⇒ (b) Let P , Q be prime ideals such that P ⊆ Q and suppose that P 6= Q. By hypothesis, there are two ideals I and J such that P ⊆ r(I), Q ⊆ r(J) and r(I) ∩ r(J) = ∅. It follows that I * P , J * Q and r(I ∩ J) = r(0). Since P ⊆ Q we infer that J * P and, by Proposition 3.6.1 (e), I ∩ J = {0}. Because {0} ⊆ P and P is prime we get I ⊆ P or J ⊆ P , which is a contradiction. Hence, P = Q and P is a maximal ideal. (b) ⇒ (a) Follows by Theorem 3.6.10.

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PROPOSITION 3.6.13. If A is an MV-algebra such that Spec(A) = Max (A), then Spec(A) is a Boolean space. Proof. Recall that any compact set of a Hausdorff space is closed and any closed set of a compact space is compact. Since Spec(A) is a compact Hausdorff space by Theorem 3.6.10, the compact open sets coincide with its clopen sets. By Proposition 3.6.4, the clopen sets {r(a) | a ∈ A} form a basis so Spec(A) is a Boolean space.

4

Classes of MV-algebras

This section deals with some important classes of MV-algebras. The MV-chains are characterized as quotients of MV-algebras with respect to prime ideals. Chang’s representation theorem (Theorem 4.1.4) asserts that any MV-algebra is a subdirect product of MV-chains. Thus the algebraic calculus in arbitrary MV-algebras is reduced to calculus in MV-chains. The subalgebras of the MV-algebra [0, 1] are characterized: they are either isomorphic with Łn for some natural number n, or dense in [0, 1]. We also investigate the simple MV-algebras (according to the classical definitions from universal algebra, these are structures with only two congruences), which are proved to be exactly the quotients with respect to maximal ideals. The semisimple structures have the property that the intersection of their maximal ideals is {0}, while the Archimedean MV-algebras are structures without infinitesimals (according to the definition of Archimedean Abelian lattice ordered groups). Hence, the semisimple and Archimedean MV-algebras coincide. Local MV-algebras are structures with only one maximal ideal. We further characterize and classify these algebras. The proper subclass of perfect MV-algebras, which are generated by their radical, is proved to be categorical equivalent with the Abelian `-groups in Section 5.5. 4.1

MV-chains

In this section we prove that any MV-algebra can be represented as a subdirect product of MV-chains. An important property says that taking the quotient of an MV-algebra with respect to a prime ideal one gets an MV-chain. Hence, the algebraic calculus can be reduced to linearly ordered structures. An identity holds in any MV-algebra if and only if it holds considering all the possible orderings for the variables involved. We also classify the subalgebras of [0, 1], which are either isomorphic with Łn for some natural number n, or dense in [0, 1]. DEFINITION 4.1.1. We say that an MV-algebra is MV-chain if it is linearly ordered. PROPOSITION 4.1.2. For a nontrivial MV-algebra A the following are equivalent: (a)

A is an MV-chain,

(b)

any proper ideal of A is prime,

(c) {0} is a prime ideal, (d)

Spec(A) is linearly ordered.

Proof. (a) ⇒ (b) Let I be a proper ideal of A and x, y ∈ A. By hypothesis, it follows that x ≤ y or y ≤ x, so x y ∗ = 0 ∈ I or y x∗ = 0 ∈ I. Thus, I is a prime ideal of A.

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(b) ⇒ (c) Straightforward. (c) ⇒ (d) By Proposition 3.3.4 and the fact that {0} is a prime ideal, we infer that any proper ideal is prime, so SpecA = {I ⊆ A | I is a proper ideal and {0} ⊆ I}. Thus Spec(A) is linearly ordered under the set theoretical inclusion (Proposition 3.3.6). (d) ⇒ (a) Let x, y ∈ A and suppose x 6≤ y and y 6≤ x, so x y ∗ 6= 0 and y x∗ 6= 0. By Proposition 3.3.11, there are P and Q prime ideals such that x y ∗ ∈ / P and y x∗ ∈ / Q. Hence, using the definition of prime ideals, we get y x∗ ∈ P and x y ∗ ∈ Q. By hypothesis, Spec(A) is linearly ordered, so P ⊆ Q or Q ⊆ P . Thus, y x∗ ∈ Q or x y ∗ ∈ P which is a contradiction. We have x ≤ y or y ≤ x, so A is an MV-chain. PROPOSITION 4.1.3. If A is an MV-algebra and I is a proper ideal of A, then the following are equivalent: (a)

I is a prime ideal,

(b)

A/I is an MV-chain. ∗

Proof. (a) ⇒ (b) If a, b ∈ A, then a b∗ ∈ I or b a∗ ∈ I, so [a]I [b]I = [0]I or ∗ [b]I [a]I = [0]I . Thus, [a]I ≤ [b]I or [b]I ≤ [a]I in A/I, i.e. A/I is an MV-chain. (b) ⇒ (a) If A/I is an MV-chain, then [a]I ≤ [b]I or [b]I ≤ [a]I for any a, b ∈ A. It follows that a b∗ ∈ I or b a∗ ∈ I for any a, b ∈ A, so I is a prime ideal. THEOREM 4.1.4 (Chang’s representation theorem). Every MV-algebra is a subdirect product of MV-chains. Proof. By Propositions 3.3.13 and 4.1.3. COROLLARY 4.1.5. If A is an MV-algebra, a, b ∈ A and n ∈ N, then the following identities hold: (a)

n(a ∨ b) = na ∨ nb,

(b)

n(a ∧ b) = na ∧ nb,

(c) b ⊕ c = c implies a b ⊕ a c ⊕ b = a c ⊕ b. Proof. (a) and (b) These identities obviously hold if A is an MV-chain. Hence, by Theorem 4.1.4, the identities hold in any MV-algebra. (c) Let b ⊕ c = c. By Theorem 4.1.4 we can assume that A is an MV-chain. Then we get b = 0 or c = 1. If b = 0 the equality a b ⊕ a c ⊕ b = a c ⊕ b is trivial. If c = 1, by Proposition 2.2.11 (d) the equality holds too. PROPOSITION 4.1.6. Every MV-chain is indecomposable. Proof. Let A be an MV-chain and a ∈ B(A). If a ≤ a∗ , then a = a ∧ a∗ = 0. If a ≥ a∗ , then a = a ∨ a∗ = 1. Thus, B(A) = {0, 1} and the desired conclusion follows by Proposition 2.8.7. PROPOSITION 4.1.7. If A is an MV-chain, then M = {a ∈ A | ord (a) = ∞} is the unique maximal ideal of A.

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Proof. Obviously, M is not empty, since 0 ∈ M . Suppose a ≤ b and b ∈ M . If a has a finite order, then so has b, which contradicts the fact that b ∈ M . Thus, ord (a) = ∞, so a ∈ M . Consider a, b ∈ M and suppose a ⊕ b ∈ / M . Therefore there is n ∈ N such that n(a ⊕ b) = 1. Since A is an MV-chain we have a ≤ b or b ≤ a. Hence n(a ⊕ b) ≤ 2na or n(a ⊕ b) ≤ 2nb, so 2na = 1 or 2nb = 1. This contradicts the fact that a and b are in M . We get a ⊕ b ∈ M , so M is a proper ideal of A. In order to prove that M is the unique maximal ideal of A, it is enough to prove that any proper ideal of A is included in M . Indeed, if I is a proper ideal of A and a ∈ I, then ord (a) = ∞, so a ∈ M . COROLLARY 4.1.8. If A is an MV-chain, then Spec(A) = Id (A) is a bounded chain with first element {0} and last element M = {a ∈ A | ord (a) = ∞}. Proof. By Propositions 4.1.2 and 4.1.7. We recall some basic notions from the theory of partially ordered sets. If A is an MV-algebra, then an element a ∈ A is an atom if a > 0 and there is no x ∈ L such that 0 < x < a. We say that A is atomless if it has no atoms. An MV-algebra A is densely ordered if the order relation of A is dense, i.e. {x ∈ A | a < x < b} = 6 ∅ whenever a < b in A. A subset X ⊆ A is dense in A if X ∩{x ∈ A | a < x < b} = 6 ∅ for any a < b in A. PROPOSITION 4.1.9. Let A be an atomless MV-algebra. Then A is densely ordered. Proof. Assume that for some a, b ∈ A, {x ∈ A | a < x < b} = ∅. Let c = b a∗ > 0. A is atomless, so for some x ∈ A, 0 < x < c. Then b x∗ ≥ b c∗ = b (b∗ ⊕ a) = a. Therefore, a ⊕ x ≤ b. This yields a ⊕ x = a or a ⊕ x = b. The former implies x = 0, in contradiction with x > 0. If a ⊕ x = b, then c = a∗ ∧ x ≤ x, absurd. PROPOSITION 4.1.10. Any MV-chain with an atom of order n is isomorphic to Łn+1 . Proof. Let A be an MV-chain and let a ∈ A be an atom such that ord (a) = n. It follows that 0 < a < 2a < · · · < (n − 1)a < na = 1. If b ∈ A, then there is k < n such that ka ≤ b ≤ (k + 1)a. It follows that b (ka)∗ ≤ (ka)∗ ∧ a ≤ a. Since a is an atom we get b (ka)∗ = 0 or b (ka)∗ = a. If b (ka)∗ = 0, then ka ≤ b ≤ ka, so b = ka. If b (ka)∗ = a, then b = b ∨ ka = b (ka)∗ ⊕ ka = a ⊕ ka = (k + 1)a. We proved that for any element b of A there is 0 ≤ k ≤ n such that b = ka, i.e. A = {ka | 0 ≤ k ≤ n}. Now, we will prove that (ka)∗ = (n − k)a for any 0 ≤ k ≤ n. Because ka ⊕ (n − k)a = na = 1 we get (ka)∗ ≤ (n − k)a. If we suppose that (ka)∗ ≤ (n − k − 1)a we have (n − 1)a = ka ⊕ (n − k − 1)a = 1, which contradicts the fact that ord (a) = n. Thus (ka)∗ = (n − k)a for any 0 ≤ k ≤ n. Since we also have ka ⊕ la = (k + l)a for any 0 ≤ k, l ≤ n, it is obvious that the function f : A → Ln+1 defined by f (ka) = k/n is an MV-algebra isomorphism. COROLLARY 4.1.11. Every finite subalgebra of [0, 1] is isomorphic to Łn for some n ≥ 2. Proof. Let A be a finite subalgebra of [0, 1] and let a = min{x ∈ A | x > 0}. It is obvious that a is an atom. Since A is a subalgebra of [0, 1], there exists n ∈ N such that ord (a) = n. If n = 1, then A = L2 . Otherwise, A has an atom of order n for some n ≥ 2. By Proposition 4.1.10, A is isomorphic to Łn+1 .

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49

COROLLARY 4.1.12. Every infinite subalgebra of [0, 1] is dense in [0, 1]. Proof. By Corollary 4.1.11 and Proposition 4.1.9. 4.2

Simple and semisimple MV-algebras

The simplicity and the semisimplicity are general notions of universal algebra. An algebra is simple if it has only two congruences and it is semisimple if the intersection of its maximal congruences contains only the identity. The MV-algebras of [0, 1]-valued functions are semisimple. In fact, any semisimple MV-algebra is isomorphic with a subalgebra of [0, 1]X for some nonempty set X, as we prove in Theorem 5.4.6. DEFINITION 4.2.1. An MV-algebra A is called semisimple if Rad (A) = {0}, which means that the intersection of the maximal ideals of A is {0}. EXAMPLE 4.2.2. For any nonempty set X the MV-algebra [0, 1]X is semisimple. Indeed, let f ∈ [0, 1]X and n ∈ N such that nf ≤ f ∗ . It follows that nf (x) ≤ 1−f (x), so (n + 1)f (x) ≤ 1 for any natural number n and for any x ∈ X. Thus, f (x) = 0 for any x ∈ X. We infer that [0, 1]X has no infinitesimals, so it is a semisimple MV-algebra. We define the Archimedean MV-algebras and we prove that they coincide with the semisimple ones. LEMMA 4.2.3. In any MV-algebra A the following are equivalent: (a)

for every a ∈ A, na ≤ a∗ for any n ∈ N implies a = 0,

(b)

for every a, b ∈ A, na ≤ b for any n ∈ N implies a b = a.

Proof. (a) ⇒ (b) Let a, b ∈ A such that na ≤ b for any n ∈ N. We get n(a ∧ b∗ ) ≤ ∗ na ≤ b ∨ a∗ , so n(a ∧ b∗ ) ≤ (a ∧ b∗ ) for any n ∈ N. By hypothesis, a ∧ b∗ = 0 so a∗ ∨ b = 1. Thus a = a (a∗ ∨ b) = (a a∗ ) ∨ (a b) = 0 ∨ (a b) = a b. (b) ⇒ (a) Let a ∈ A such that na ≤ a∗ for any n ∈ N. By hypothesis we get a a∗ = a, so a = 0. DEFINITION 4.2.4. An MV-algebra A is called Archimedean if the equivalent conditions from Lemma 4.2.3 are satisfied. One can easily see that an MV-algebra is Archimedean iff it has no infinitesimals. PROPOSITION 4.2.5. An MV-algebra A is semisimple iff it is Archimedean. Proof. An MV-algebra A is semisimple iff it contains no infinitesimals. Thus, if na ≤ a∗ for any n ∈ N , then a = 0. This means that A is Archimedean. PROPOSITION 4.2.6. Any σ-complete (complete) MV-algebra is semisimple. Proof. Let A be W a σ-complete MV-algebra and a ∈ A such that na ≤ a∗ for any n ∈ N. If b = {na | n ∈ N}, then b ≤ a∗ and b ⊕ a = b. Hence a ≤ b∗ and a ∧ b∗ = b∗ (b ⊕ a) = b∗ b = 0, so a = 0. Thus, A is Archimedean, so it is semisimple.

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PROPOSITION 4.2.7. Any MV-algebra A which is not semisimple contains a subalgebra isomorphic to Chang’s algebra C. Proof. By Proposition 3.5.6, every MV-algebra A which is not semisimple contains an infinitesimal. The desired result follows by Proposition 3.5.10. DEFINITION 4.2.8. An MV-algebra A is called simple if the only ideals are {0} and A. PROPOSITION 4.2.9. For an MV-algebra A the following are equivalent: (a)

A is simple,

(b)

ord (a) < ∞ for any a ∈ A \ {0},

(c) {0} is a maximal ideal, (d)

A is linearly ordered and semisimple.

Proof. (a) ⇒ (b) Let a ∈ A, a 6= 0 and suppose ord (a) = ∞. Then the ideal (a] is proper and (a] 6= {0} which contradicts the hypothesis. (b) ⇒ (c) Let I be a proper ideal of A and a ∈ I. If a 6= 0, then na = 1 for some n ∈ N. Because na ∈ I we get a contradiction. Thus a = 0 so {0} is the only proper ideal of A. Obviously, {0} is the only maximal ideal. (c) ⇒ (a) Obvious. (c) ⇒ (d) Since {0} is a maximal ideal it is also prime so, by Proposition 4.1.2, A is an MV-chain. It is obvious that the intersection of the maximal ideals of A is {0}. Hence, A is semisimple. (d) ⇒ (b) Let a be a nonzero element from A. Since A is Archimedean, there is n ∈ N such that na 6≤ a∗ . It follows that a∗ ≤ na so a∗ ⊕ a ≤ (n + 1)a, hence ord (a) = n + 1 < ∞. PROPOSITION 4.2.10. The following are equivalent for any proper ideal M of A: (a)

M is maximal,

(b)

A/M is simple.

Proof. (a) ⇒ (b) Let a be in A such that [a]M 6= [0]M , so a ∈ / M . By Proposi∗ tion 3.4.2, there is n ∈ N such that (a∗ )n ∈ M . Hence ([a]M )n = [0]M and we get n[a]M = [1]M . We proved that every nonzero element from A/M has a finite order so, by Proposition 4.2.9, A/M is a simple MV-algebra. (b) ⇒ (a) Let I be an ideal of A such that M ⊆ I and consider a ∈ I \ M . Because [a]M 6= [0]M , by Proposition 4.2.9, there is n ∈ N such that n[a]M = [1]M , so [na]M = ∗ ∗ [1]M . Thus, (na) = d(na, 1) ∈ M ⊆ I. We get na, (na) ∈ I, so I = A. Hence M is maximal. PROPOSITION 4.2.11. (a)

Any simple MV-algebra is an MV-chain.

(b)

Any MV-chain which is not simple has a subalgebra isomorphic to Chang’s algebra C.

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Proof. (a) Let a, b ∈ A such that a 6≤ b, i.e. a b∗ 6= 0. Then there is a natural number n such that n(a b∗ ) = 1. By Propositions 2.2.11 and 2.2.12 we get n(a∗ b) = 0 so a∗ b = 0, i.e. b ≤ a. Thus A is linearly ordered. (b) Suppose A is an MV-chain which is not simple. Then there is a nonzero element a ∈ A such that ord (a) = ∞. We will prove that a is an infinitesimal. Suppose that there is n ∈ N such that na 6≤ a∗ . Since A is linearly ordered, we get a∗ ≤ na, so (n + 1)a = 1 which is a contradiction. Thus, na ≤ a∗ for any n ∈ N, so a is an infinitesimal. The desired result follows by Proposition 3.5.10. THEOREM 4.2.12. Every semisimple MV-algebra is a subdirect product of simple MV-algebras. Proof. By Proposition 3.2.11, 3.4.2, and 4.2.9. FACT 4.2.13. In Section 5 we will prove that the simple MV-algebras are, up to isomorphism, the subalgebras of [0, 1]. Thus, by Proposition 4.2.12 any semisimple MV-algebra is a subdirect product of subalgebras of [0, 1]. In other words, for any semisimple MV-algebra A there is a set X such that A is isomorphic to a subalgebra of [0, 1]X (the set X is in fact Max (A)). Using the terminology from [4], the subalgebras of [0, 1]-valued functions are called bold algebras of fuzzy sets. Hence, by Example 4.2.2 and Proposition 4.2.12, we conclude that the semisimple MV-algebras are, up to isomorphism, the bold algebras of fuzzy sets. 4.3

Local and perfect MV-algebras

PROPOSITION 4.3.1. For any MV-algebra A, the following are equivalent: (a)

for any a ∈ A, ord (a) < ∞ or ord (a∗ ) < ∞,

(b)

for any a, b ∈ A, a b = 0 implies an = 0 or bn = 0 for some n ∈ N,

(c)

for any a, b ∈ A, ord (a ⊕ b) < ∞ implies ord (a) < ∞ or ord (b) < ∞,

(d)

{a ∈ A | ord (a) = ∞} is a proper ideal of A,

(e) A has only one maximal ideal. Proof. (a) ⇒ (b) Let a, b ∈ A such that a b = 0 and ak 6= 0 for any k ∈ N. Then k(a∗ ) 6= 1 for any k ∈ N, so ord (a∗ ) = ∞. By (a) it follows that ord (a) < ∞, so there exists n ∈ N such that na = 1. But a b = 0 implies that a < b∗ . Hence n(b∗ ) = 1, which means that bn = 0. (b) ⇒ (c) If a, b ∈ A such that ord (a ⊕ b) < ∞, then there exists some n ∈ N such that n(a ⊕ b) = 1. It follows that (a∗ b∗ )n = 0 so, by hypothesis, (a∗ )n = 0 or (b∗ )n = 0. Hence na = 1 or nb = 1 and the desired conclusion is obvious. (c) ⇒ (d) Let M = {a ∈ A | ord (a) = ∞} and a < b ∈ A with b ∈ M . If we assume that ord (a) < ∞, then na = 1 for some n ∈ N, so nb = 1 which is a contradiction. Hence ord (a) = ∞ and a ∈ M . Let a, b ∈ M , so ord (a) = ord (b) = ∞. Assume that ord (x ⊕ y) < ∞. By hypothesis it follows that ord (a) < ∞ or ord (b) = ∞, so a ∈ /M or b ∈ / M . But this is a contradiction. We proved that a ⊕ b ∈ M , so M is an ideal of A. It is obvious that I is a proper ideal since 1 ∈ / M.

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(d) ⇒ (e) If J is a proper ideal of A, then ord (x) = ∞ for every x ∈ J. Hence J ⊆ M = {a ∈ A | ord (a) = ∞} for any proper ideal J of A. It is straightforward that M is the only maximal ideal of A. (e) ⇒ (a) Let M be the only maximal ideal of A. By Remark 2.7.2 (i8) and Proposition 3.4.5 it follows that, for any a ∈ A, ord (a) = ∞ implies that a ∈ M . Hence if we assume that ord (a) = ord (a∗ ) = ∞ for some a ∈ A, then a, a∗ ∈ M and M is not a proper ideal. In consequence, ord (a) < ∞ or ord (a∗ ) < ∞ for any a ∈ A. DEFINITION 4.3.2. An MV-algebra A is called local if the equivalent conditions from Proposition 4.3.1 are satisfied. FACT 4.3.3. If A is a local MV-algebra, then Rad (A) = {a ∈ A | ord (A) = ∞} is the only maximal ideal of A. PROPOSITION 4.3.4. For an MV-algebra A and a proper ideal P ⊆ A, the following are equivalent: (a)

there is a unique maximal ideal containing P ,

(b)

A/P is a local MV-algebra,

(c) a b ∈ P implies an ∈ P or bn ∈ P for some integer n, (d)

for any a ∈ A there exists some n ∈ N such that an ∈ P or (a∗ )n ∈ P .

Proof. (a) ⇒ (b) By Remark 3.4.3, the MV-algebra A/P has only one maximal ideal, so A/P is a local MV-algebra. (b) ⇒ (c) Recall that for any a ∈ A, a ∈ P iff [a]P = [0]P . Our conclusion follows by Proposition 4.3.1 (b). (c) ⇒ (d) Obvious, since a a∗ = 0 ∈ P . (d) ⇒ (a) Using Remark 3.4.3 we only have to prove that the MV-algebra A/P has a unique maximal ideal. But this follows by Proposition 4.3.1. DEFINITION 4.3.5. An ideal P of an MV-algebra A is called primary if P is proper and there is a unique maximal ideal containing it. COROLLARY 4.3.6. In an MV-algebra any prime ideal is a primary ideal. Proof. It follows by Proposition 3.4.5. EXAMPLE 4.3.7. If A is an MV-algebra and P a prime ideal of A, then \ OP := {Q | Q ⊆ P, Q ∈ Spec(A)} is a primary ideal of A. It is trivial to check that OP is an ideal of A. Let MP be the unique maximal ideal of A including P . Assume there is a maximal ideal M such that M 6= MP and OP ⊆ M . Then there are elements a and b such that a ∈ M , b ∈ MP and a ⊕ b = 1. Hence we have a∗ b∗ = 0 and (a∗ )2 ∧ (b∗ )2 = 0. So for every prime ideal Q ⊆ P we must have either (a∗ )2 ∈ Q or (b∗ )2 ∈ Q. If (b∗ )2 ∈ Q, then (b∗ )2 ∈ MP . Hence b ∨ b∗ = b ⊕ (b∗ )2 ∈ MP , so b∗ ∈ MP which is absurd. So (a∗ )2 ∈ Q for any prime ideal Q ⊆ P . It follows that (a∗ )2 ∈ OP ⊆ M . Similarly as above, we get a ∨ a∗ = a ⊕ (a∗ )2 ∈ M which is absurd. So, the ideal OP is included in only one maximal ideal. We proved that OP is a primary ideal.

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PROPOSITION 4.3.8. For an MV-algebra A, the following are equivalent: (a)

A is local,

(b)

any proper ideal of A is primary,

(c) {0} is a primary ideal, (d)

Rad (A) contains a primary ideal.

Proof. (a) ⇒ (b) Let P be a proper ideal of A. By Proposition 4.3.1, A has only one maximal ideal. Hence there exists a unique maximal ideal containing P . By Proposition 4.3.4, P is a primary ideal. (b) ⇒ (c) Straightforward. (c) ⇒ (d) Obvious, since {0} ⊆ Rad (A). (d) ⇒ (a) Let P be a primary ideal such that P ⊆ Rad (A). By Proposition 4.3.4, there exists a unique maximal ideal containing P . Hence Rad (A) is the only maximal ideal of A and, by Proposition 4.3.1, A is a local MV-algebra. COROLLARY 4.3.9. Let A be a local MV-algebra. Then every homomorphic image of A is local. Proof. By Theorem 3.2.8, we only have to prove that A/I is local for any proper ideal I of A. Since A is local then any proper ideal I is primary by Proposition 4.3.8. Hence, by Proposition 4.3.4, A/I is a local MV-algebra for any proper ideal I of A. PROPOSITION 4.3.10. For a local MV-algebra A the following properties hold: (a)

B(A) = {0, 1},

(b)

A is indecomposable,

(c) Spec(A) is connected. Proof. (a) Let a be a Boolean element of A. By Proposition 4.3.1 (b), there exists some n ∈ N such that an = 0 or (a∗ )n = 0. But this implies that a = 0 or a∗ = 0, so B(A) = {0, 1}. (b) Follows by (a) and Proposition 2.8.7. (c) Follows by (a) and Theorem 3.6.6. PROPOSITION 4.3.11. Let A be an MV-algebra, P a prime ideal of A and a b ∈ P . Then a2 ∈ P or b2 ∈ P . Proof. P is prime so a b∗ ∈ P or a∗ b ∈ P . We can assume a b∗ ∈ P . Thus ((a b) ⊕ (a b∗ )) ∈ P . By Proposition 2.2.4 (g) we have a (b ⊕ (a b∗ )) ∈ P . Since a (b ⊕ (a b∗ )) = a (a ∨ b) = a2 ∨ a b, it follows that a2 ∨ a b ∈ P , so a2 ∈ P . If a∗ b ∈ P one can similarly prove that b2 ∈ P . COROLLARY 4.3.12. Any MV-chain is a local MV-algebra. Proof. If A is an MV-chain then, by Proposition 4.1.2, {0} is a prime ideal. Using Corollary 4.3.6 and Proposition 4.3.4 it follows that A = A/{0} is a local MV-algebra.

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The converse of the above proposition is false. To see this, let C be the Chang algebra and let A the subalgebra of C × C having as support the following set: {hnc, mci | n, m > 0} ∪ {h1 − nc, 1 − mci | n, m > 0}. Note the MV-algebra defined above is an example of a local MV-algebra that is not an MV-chain. Moreover the ideal {h0, 0i} = 0 satisfies a b = 0 implies a2 = 0 or b2 = 0, but it is not a prime ideal. The following proposition tells us which local MV-algebras are semisimple. PROPOSITION 4.3.13. Let A be a local MV-algebra. If A is semisimple, then A is simple. Proof. If A is local then, by Remark 4.3.3, Rad (A) is the only maximal ideal of A. Since A is also semisimple, Rad (A) = {0}, so {0} is a maximal ideal. By Proposition 4.2.9 it follows that A is a simple MV-algebra. LEMMA 4.3.14. Let A be a local MV-algebra and ρ : A → [0, 1] a homomorphism. Then ker (ρ) = Rad (A). Proof. By Proposition 4.2.10, ρ−1 (0) is a maximal ideal of A. Since A is local it has only one maximal ideal, then ker (ρ) = ρ−1 (0) = Rad (A). The following result is a topological characterization for local MV-algebras. THEOREM 4.3.15. For an MV-algebra A the following are equivalent: (a)

A is local,

(b)

Spec(A) is connected and Max (A) is closed in Spec(A).

Proof. (a) ⇒ (b) Assume that A is a local MV-algebra. Then, by Proposition 4.3.10, Spec(A) is connected. Let M be the unique maximal ideal of A. Then Max (A) = {M } = Spec(A) \ r(M ), so Max (A) is closed. (b) ⇒ (a) Since Max (A) is closed, by Proposition 3.6.7 we have Rad (A) ⊆ P

iff P ∈ Max (A)

for any P ∈ Spec(A). For every Q0 ∈ Spec(A/Rad (A)) there is Q ∈ Spec(A) such that Rad (A) ⊆ Q and Q0 = Q/Rad (A). It follows that Q ∈ Max (A) and Q0 ∈ Max (A/Rad (A)). So we get Spec(A/Rad (A)) = Max (A/Rad (A)). Thus, by Proposition 3.6.13 Spec(A/Rad (A)) is a Boolean space. Assume that A is not local. Hence Spec(A/Rad (A)) is a non trivial Boolean space, hence is not connected. Now, each prime ideal P ∈ Spec(A) is contained in a unique maximal ideal, thus we have a retraction from Spec(A) to Max (A) given by M : P → MP . The map M is continuous. Since Spec(A) is connected, then also Max (A) is connected. We also have a bijection τ : M → M/Rad (A) from Max (A) to Spec(A/Rad (A)). It is straightforward to check that τ is continuous. Since Max (A) is compact and Spec(A/Rad (A)) is Hausdorff, we can infer that Max (A) is homeomorphic to Spec(A/Rad (A)). Thus Spec(A/Rad (A)) is connected. This contradiction shows that A must be local.

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Let A be an MV-algebra. We recall that an element a of A is called finite iff ord (a) < ∞ and ord (a∗ ) < ∞. Denote Fin(A) the set of all finite elements of A. PROPOSITION 4.3.16. For an MV-algebra A the following are equivalent: (a)

A is local,

(b)

A = hRad (A)i ∪ Fin(A) = Rad (A) ∪ Rad (A)∗ ∪ Fin(A).

Proof. Note that hRad (A)i = Rad (A) ∪ Rad (A)∗ by Proposition 2.7.5. (a) ⇒ (b) Assume A to be local and x ∈ A \ hRad (A)i. Then [a]Rad(A) > 0 and [a∗ ]Rad(A) > 0. Since A/Rad (A) is simple, then there are integers n, m such that [na]Rad(A) = 1 and [ma∗ ]Rad(A) = 1. Hence (na)∗ ∈ Rad (A) and (ma∗ )∗ ∈ Rad (A), i.e., na ∈ (Rad (A))∗ and ma∗ ∈ (Rad (A))∗ . So, 2na = 1 and 2ma∗ = 1. That is ord (a) < ∞ and ord (a∗ ) < ∞. Hence a is finite. (b) ⇒ (a) For a ∈ hRad (A)i we have either ord (a) < ∞ or ord (a∗ ) < ∞. Of course, if a is finite we still have either ord (a) < ∞ or ord (x∗ ) < ∞. EXAMPLE 4.3.17. Let R be the Abelian `-group of real numbers and G an Abelian `-group. Consider r0 > 0 in R and g0 ≥ 0 in G. If H = R ×lex G is the lexicographic product of R and G, then hr0 , g0 i > 0 in H. One can easily see that [h0, 0i, hr0 , g0 i] = {h0, gi | g ∈ G, g ≥ 0} ∪ {hr0 , gi | g ∈ G, g ≤ g0 }∪ {hr, gi | r ∈ R, 0 < r < r0 , g ∈ G}. Consider now the MV-algebra A = [h0, 0i, hr0 , g0 i]H defined as in Lemma 2.5.1. Note that for every real number r > 0 there exists n ∈ N such that nr > r0 . Hence, for any hr, gi ∈ A, ord (hr, gi) = ∞ iff r = 0. One can prove that Rad (A) = {h0, gi | g ∈ G, g ≥ 0}, Rad (A)∗ = {hr0 , gi | g ∈ G, g ≤ g0 }, Fin(A) = {(r, g) | r ∈ R, 0 < r < r0 , g ∈ G}. Therefore A is a local MV-algebra. Moreover, A is an MV-chain iff G is a totally ordered group. PROPOSITION 4.3.18. Let A be a local MV-algebra. Then for every a, b ∈ A such that [a]Rad(A) 6= [b]Rad(A) , either a < b or b < a. Proof. By hypothesis we either have a∗ b ∈ / Rad (A) or b∗ a ∈ / Rad (A). In the ∗ first case, since A is local, then we have ord (a b) < ∞ and then by Corollary 2.2.14, a∗ ⊕ b = 1, i.e., a < b. In the second case, from b∗ a ∈ / Rad (A) we similarly obtain b < a. PROPOSITION 4.3.19. For an MV-algebra A the following are equivalent: (a)

A is local,

(b)

For every x ∈ A, x ≤ x∗ or x∗ ≤ x or (d(x, x∗ ))2 = 0.

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Proof. (a) ⇒ (b) For any x ∈ A, if x ≡ x∗ does not hold, we have either x < x∗ or x∗ < x. In the case that x ≡ x∗ we have d(x, x∗ ) ∈ Rad (A), and then d(x, x∗ )2 = 0. (b) ⇒ (a) If x ≤ x∗ , then ord (x∗ ) < ∞. Analogously, if x∗ ≤ x, then ord (x) < ∞. If d(x, x∗ )2 = 0, i.e., (x2 ⊕ (x∗ )2 )2 = 0, then for every prime ideal P of A we have the following cases: (i)

[x]P ≤ [x∗ ]P ,

(ii)

[x∗ ]P ≤ [x]P .

Assuming (i) we get [x]P [x]P = 0, and then ([x∗ ]P )4 ) = 0. Hence ord ([x]P ) ≤ 4. While, assuming (ii) holds we get ord (x) ≤ 2. Hence, for every prime ideal P of A, ord ([x]P ) ≤ 4. This implies that ord (x) < ∞. Hence A is local. COROLLARY 4.3.20. The class of local MV-algebras is a universal class. Proof. By Proposition 4.3.19. Let us describe an example of local MV-algebra that will result as a kind of prototypical local MV-algebra. EXAMPLE 4.3.21. Let X be an arbitrary nonempty set, U a ultrapower of the MValgebra [0, 1], and K(U X ) the subset of the MV-algebra U X as follows: K(U X ) = {f ∈ U X | f (X) ⊆ [a]Rad(U )

for some a ∈ U }.

The algebra K(U X ) will be called the the full MV-algebra of quasi constant functions from X to U . Of course any element f from K(U X ) will be said quasi constant function from X to U . Any subalgebra of K(U X ) will be called an algebra of quasi constant functions. PROPOSITION 4.3.22. K(U X ) is a local MV-algebra. Proof. Let us show that K(U X ) is a subalgebra of U X . The zero constant function f0 belongs to K(U X ) because f0 (X) ⊆ Rad (U ) = [0]Rad(U ) . Similarly, from f1 (X) ⊆ (Rad (U ))∗ = [1]Rad(U ) we have f1 ∈ K(U X ). Assume f satisfies f (X) ⊆ [a]Rad(U ) for some a ∈ U . Then we have f ∗ (X) ⊆ [a∗ ]Rad(U ) . Finally, let f, g ∈ K(U X ) be such that f (X) ⊆ [a]Rad(U ) for a ∈ U and g(X) ⊆ [b]Rad(U ) for b ∈ U . Then (f ⊕ g)(X) ⊆ [a ⊕ b]Rad(U ) . Hence K(U X ) is a subalgebra of U X . To show that K(U X ) is local take f ∈ K(U X ). If f (X) ⊆ [0]Rad(U ) , then f ∗ (X) ⊆ [1]Rad(U ) and ord (f ∗ ) < ∞. If f (X) ⊆ [1]Rad(U ) we have ord (f ) < ∞. Assume now that f (X) ⊆ [a]Rad(U ) 6= [0]Rad(U ) 6= [1]Rad(U ) . Then, for every x ∈ X, f (x) ∼Rad(U ) a and a ∈ / Rad (U ). Then ord (f ) < ∞ and ord (f ∗ ) < ∞. Given a local MV-algebra A we know the quotient algebra A/Rad (A) to be simple. DEFINITION 4.3.23. Let n be a positive integer. Then a local MV-algebra A is said to be of rank n iff A/Rad (A) ' Łn+1 . An MV-algebra A is said to be of finite rank iff A is a local MV-algebra of rank n for some integer n.

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PROPOSITION 4.3.24. If A is a local MV-algebra of rank n, then there exists an element b in A such that the following hold: (a)

{0, b, 2b, . . . , nb} ∩ [x]Rad(A) is a singleton for every x ∈ A,

(b)

(rb)∗ = sb ⊕ ((r + s)b)∗ and rb ⊕ sb = (r + s)b for every r, s = 1, . . . , n with r + s ≤ n.

Proof. (a) Let ψ be the isomorphism from A/Rad (A) onto Łn+1 and take any a ∈ ψ −1 (1/n). Hence na ∈ ψ −1 (1) = Rad ∗ (A) and 1 6= (n − 1)a ≤ na ≤ 1. If na < 1, then a > (na)∗ and hence (n+1)a = na⊕a > na⊕(na)∗ = 1, and so ord (a) = n+1. Note that both a and ((n − 1)a)∗ ∈ ψ −1 (1/n). Take b = a ∧ ((n − 1)a)∗ ∈ ψ −1 (1/n). Hence (a) is satisfied. (b) One can prove, by induction on r = 1, . . . , n, that (n − r)b ≤ (rb)∗ . Hence, we have: sb ⊕ ((r + s)b)∗ = sb ⊕ ((rb ⊕ sb))∗ = sb ⊕ ((rb)∗ (sb)∗ ) = sb ∨ (rb)∗ where we applied the definition x ∨ y = x ⊕ (x∗ y). Since sb ≤ (n − r)b ≤ (rb)∗ we get (rb)∗ = sb ⊕ ((r + s)b)∗ . Hence rb = (sb)∗ (r + s)b and so rb ⊕ sb = ((sb)∗ (r + s)b) ⊕ sb = sb ∨ (r + s)b and since sb ≤ (r + s)b we get the claim. MV-algebras of finite rank are used in Section 7 and they are characterized in Theorem 5.5.6. In the following we will classify the local MV-algebras. DEFINITION 4.3.25. An MV-algebra A is called perfect if for any a ∈ A, ord (a) = ∞ iff ord (a∗ ) < ∞. It is straightforward that any perfect MV-algebra is local. EXAMPLE 4.3.26. Chang’s MV-algebra C from Example 2.4.5 is a perfect MValgebra. DEFINITION 4.3.27. An MV-algebra A is called singular if A is local and there exist a, b ∈ A such that ord (a) < ∞, ord (b) < ∞ and a b ∈ Rad (A) \ {0}. EXAMPLE 4.3.28. The MV-algebra ∗ [0, 1] from Example 2.5.3 is a singular MValgebra. Since ∗ [0, 1] is an MV-chain it follows that it is also local. Let a = [1/2] + τ , where τ is an infinitesimal. Then ord (a) = 2 < ∞ and a a = 2τ ∈ Rad (∗ [0, 1])\{0}. LEMMA 4.3.29. Let A be a local MV-algebra which is not singular. If S = (A \ hRad (A)i) ∪ {0, 1}, then: (a)

S is an MV-subalgebra of A,

(b)

S is simple,

(c) S ' A/Rad (A).

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Proof. (a) It is obvious that a ∈ S implies a∗ ∈ S. Since 0, 1 ∈ S it suffices to prove that S is closed with respect to the operation. If a, b ∈ S \ {0} then, by Remark 4.3.4, ord (a) < ∞ and ord (b) < ∞. Assume that a b ∈ / S. It follows that a b ∈ Rad (A) \ {0}, so A is singular. But this fact contradicts our hypothesis, so a b ∈ S. (b) Follows by Remark 4.3.3. (c) Let h : S → A/Rad (A) be defined by h(s) = [s]Rad(A) for any s ∈ S. It is obvious that h is an MV-algebra homomorphism. If h(s) = [0]Rad(A) , then s ∈ Rad (A)∩S, so s = 0. Thus h is an injective homomorphism. Let [x]Rad(A) ∈ A/Rad (A) such that [x]Rad(A) 6= [0]Rad(A) and [x]Rad(A) 6= [1]Rad(A) . Hence x ∈ / hRad (A)i, so x ∈ S and h(x) = [x]Rad(A) . We proved that h is surjective, so h is an MV-algebra isomorphism. The following theorem provides a classification of local MV-algebras. Note that the only local MV-algebra which is a Boolean algebra is Ł2 = {0, 1}. THEOREM 4.3.30. For any local MV-algebra A if A 6= Ł2 , then exactly one of the following holds: (a)

A is perfect,

(b)

A is singular,

(c) A is simple. Proof. Assume that A is a local MV-algebra and A 6= Ł2 . It is obvious that A cannot be perfect and simple simultaneously and A cannot be singular and simple simultaneously. If A is singular, there are a, b ∈ A such that ord (a) < ∞, ord (b) < ∞ and a b ∈ Rad (A) \ {0}. If A is also perfect, then ord (a∗ ) = ord (b∗ ) = ∞, so a∗, b∗ ∈ Rad (A). ∗ Hence (a b) = a∗ ⊕ b∗ ∈ Rad (A), which is impossible because Rad (A) is a proper ideal of A. We proved that A cannot be singular and perfect simultaneously. Assume now that A it neither perfect, nor singular and let S = (A\hRad (A)i)∪{0, 1}. We have to prove that A is simple, i.e. Rad (A) = {0}. Let a ∈ Rad (A) be an arbitrary element. Since A is not perfect, it follows that there exists s ∈ S such that s 6= 0 and s 6= 1. Then s∗ ∈ S and ord (s∗ ) < ∞, so ord (a ⊕ s∗ ) < ∞. Hence a ⊕ s∗ ∈ / Rad (A). If we assume that a ⊕ s∗ ∈ Rad (A)∗ , then s a∗ ∈ Rad (A) and a ∨ s = a ⊕ s a∗ ∈ Rad (A). It follows that s ∈ Rad (A), which is impossible. We proved that a ⊕ s∗ ∈ / hRad (A)i, so a⊕s∗ ∈ S. Hence s∧a = s (a⊕s∗ ) ∈ S. But a ∈ Rad (A), so s∧a ∈ Rad (A)∩S = {0}. We get s ∧ a = 0. Since s ∈ S and s 6= 0 it follows that ord (s) = ∞, so there exists some n ∈ N such that ns = 1. By Corollary 4.1.5, n(s ∧ a) = ns ∧ na and we infer that na = 0. It follows that a = 0 and Rad (A) = {0}, so A is a simple MV-algebra. EXAMPLE 4.3.31. Let G be an `-group, then [h0, 0i, h2, 0i]Z×lex G is a local MValgebra which is singular and not totally ordered. We further characterize the perfect MV-algebras.

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PROPOSITION 4.3.32. For an MV-algebra A, the following are equivalent: (a)

A is perfect,

(b)

A = hRad (A)i = Rad (A) ∪ Rad (A)∗ ,

(c) A/Rad (A) ' Ł2 . Proof. (a) ⇒ (b) tion 4.3.16.

If A is perfect, then Fin(A) = ∅. Hence A = hRad (A)i by Proposi-

(b) ⇒ (c) For any x ∈ A we have x ∈ Rad (A) or x ∈ Rad (A)∗ . Hence [x]Rad(A) = [0]Rad(A) or [x]Rad(A) = [1]Rad(A) and the conclusion follows. (c) ⇒ (a) If x ∈ A, then [x]Rad(A) = [0]Rad(A) or [x]Rad(A) = [1]Rad(A) . Thus we get x ∈ Rad (A) or x ∈ Rad (A)∗ . If x ∈ Rad (A), then ord (x) = ∞ and so ord (x∗ ) = 2 < ∞ by Lemma 3.5.2 (a). If x ∈ Rad (A)∗ , then ord (x∗ ) = ∞ and ord (x) = 2 < ∞. We proved that A is perfect. EXAMPLE 4.3.33. Let Z be the Abelian `-group of integers and G an Abelian `-group. Note that h1, 0i is a strong unit of the lexicographic product Z ×lex G and [h0, 0i, h1, 0i] = {h0, gi | g ∈ G, g ≥ 0} ∪ {h1, gi | g ∈ G, g ≤ 0}. If we denote ∆(G) = [h0, 0i, h1, 0i]Z×lex G , the interval MV-algebra defined as in Lemma 2.5.1, then h0, gi∗ = h1, −gi for any g ≥ 0

and

h1, gi∗ = h0, −gi for any g ≤ 0.

One can easily prove that ord (h0, gi) = ∞ for every g ≥ 0 and ord (h1, gi) = 2 for every g ≤ 0. Thus ∆(G) is a perfect MV-algebra and Rad (∆(G)) = {h0, gi | g ∈ G, g ≥ 0}. In Section 5.5 we will prove that for any perfect MV-algebra A there exists an Abelian `-group G such that A and ∆(G) are isomorphic MV-algebras. Moreover, such an `-group is unique up to isomorphism. FACT 4.3.34. One can easily prove that the MV-algebra ∆(Z) is isomorphic with Chang’s MV-algebra C and the isomorphism is h0, ni 7→ nc

and

h1, −ni 7→ 1 − nc

for any n ∈ N. FACT 4.3.35. The class of perfect MV-algebras is obviously closed to subalgebras and is closed under homomorphic images. Still the class of perfect MV-algebras is not equational, since it is not closed under direct products. To see this, let A be any perfect MV-algebra. By Corollary 3.5.9, Rad (A × A) = Rad (A) × Rad (A) and it is obvious that h1, 0i ∈ / Rad (A × A) ∪ Rad (A × A)∗ . Hence A × A is not a perfect MV-algebra. In Section 5.5 we will prove that the equational class generated by perfect MV-algebras is in fact generated by Chang’s MV-algebra C.

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MV-algebras and Abelian `u-groups

This section contains some of the most important results from the theory of MValgebras. The first one is Mundici’s categorical equivalence between the category of MV-algebras and the category of Abelian `-groups with strong unit [20, 59]. This result led to a considerable development of the domain. The correspondence between MV-chains and totally ordered Abelian `-groups with strong unit was established by Chang in [13]. His goal was an algebraic proof for the completeness of Łukasiewicz’s ∞-valued logic. Chang’s proof of the completeness theorem is presented in this section. The last subsection contains Di Nola’s representation theorem [21]. The representation of the MV-algebras as subdirect product of chains is very useful in practice, but it offers few information about their structure. Di Nola’s representation theorem asserts that any MV-algebra is isomorphic to an algebra of nonstandard real valued functions. 5.1

The functor Γ

Recall that MV is the category of MV-algebras. The objects of MV are MValgebras and the morphisms are the MV-algebra homomorphisms. We will denote by ALGu the category of `u-groups. The elements of this category are pairs hG, ui where G is an Abelian `-group and u is a strong unit of G. The morphisms will be `-group homomorphisms which preserve the strong unit. This means that h : hG, ui → hH, vi is an `u-group homomorphism if h : G → H is an `-groups homomorphism and h(u) = v. In order to prove the categorical equivalence we define two functors: Γ : ALGu → MV and

Ξ : MV → ALGu.

The definition of the functor Γ is straightforward: Γ(G, u) := [0, u]G if hG, ui is an `u-group, Γ(h) := h|[0,u] if h : hG, ui → hH, viis an `u-group homomorphism. It is more difficult to define the functor Ξ, i.e. given an MV-algebra A to construct an `u-group GA with a strong unit u such that A is isomorphic with [0, u]GA . Then we have to prove that an `u-group G with strong unit u is isomorphic with G[0,u]G . For any MV-algebra A, the group GA is usually called the Chang’s group of A because, in the particular case when A is an MV-chain, the `-group GA was firstly defined by Chang [13]. The construction in the general case and the categorical equivalence between MValgebras and `u-groups are due to Mundici [59]. An alternative construction for GA is given in [22] using the notion of clan. If A is an MV-algebra and 1 is its greatest element, then the MV-algebras isomorphism between A and [0, 1]GA follows by an easy computation. If G is an `u-group with a strong unit u, then the `u-groups isomorphism between G and G[0,u]G follows by the fact that [0, u] generates G.

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DEFINITION 5.1.1. For an MV-chain A we define GA as the set of all the ordered pairs hm, ai with m ∈ Z and a ∈ A. If on GA we define hm + 1, 0i = hm, 1i, ( hm + n, a ⊕ bi hm, ai + hn, bi = hm + n + 1, a bi

if a ⊕ b < 1, if a ⊕ b = 1,

−hm, ai = h−m − 1, a∗ i, then hGA , +, h0, 0ii is a group. Moreover if we set hm, ai ≤ hn, bi iff either m < n or

m = n and a ≤ b,

then GA becomes an `-group, h0, 1i is a strong unit and A is isomorphic with the MValgebra [h0, 0i, h0, 1i]GA . In the sequel, A is an MV-algebra. DEFINITION 5.1.2. A sequence a = {ak | k ∈ Z} in A is good if the following properties are satisfied: (1)

ak ⊕ ak+1 = ak for all k ∈ Z,

(2)

there is some n ∈ N such that ak = 0 for all k ≥ n and ak = 1 for all k < −n.

Let GA be the set of all the good sequences in A. On GA we define the group operations as follows. DEFINITION 5.1.3. Let a = {ak | k ∈ Z} and b = {bk | k ∈ Z} be two good sequences in A. The group operations, a + b and −a, are given by M ∗ (a + b)k := (ai bj ) and (−a)k := (a−k−1 ) for all k ∈ Z. i+j=k−1

Some particular sequences will play a special role, so we introduce the following notations: NOTATION 5.1.4. If m ∈ Z and a ∈ A we will denote by hm, ai the sequence defined as follows:    1 if k < m, a if k = m, hm, aik :=   0 if k > m. m

Thus hm, ai = . . . , 1, 1, a, 0, 0, . . . We also introduce the particular notations: o := h0, 0i and u := h0, 1i.

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LEMMA 5.1.5. If a = hm, ai and b = hn, bi are good sequences in A, then the group operations from Definition 5.1.3 are given by  1 if k < m + n,      a ⊕ b if k = m + n, (1) (hm, ai + hn, bi)k = a b if k = m + n + 1,      0 if k > m + n + 1, (2)

−hm, ai = h−m − 1, a∗ i.

Proof. (1)

If we denote a = hm, ai and b = hm, bi and c = a + b, then M ck = (ai bj ) , i+j=k−1

for every k ∈ Z. Suppose k, i, j ∈ Z such that i + j = k − 1. Case 1: k > m + n + 1 It follows that i > m or j > n so ai bj = 0. Thus ck = 0. Case 2: k < m + n If i = m − 1, then j = k − 1 − i < n so ai bj = 1. It follows that ck = 1. Case 3: k = m + n If i > m, then ai bj = 0. If i < m − 1, then j = k − 1 − i > n so ai bj = 0. If i = m − 1, then j = n and ai bj = b. If i = m, then j = n − 1 and ai bj = a. It follows that cm+n = a ⊕ b. Case 4: k = m + n + 1 If i > m, then ai bj = 0. If i < m, then j > n so ai bj = 0. Thus ck = am bn = a b. The above considerations led us to the desired conclusion. (2) Straightforward. PROPOSITION 5.1.6. If A is an MV-chain, then: the good sequences in A have the form hm, ai with m ∈ Z and a ∈ A,  hm + n, a ⊕ bi if a ⊕ b < 1, (b) hm, ai + hn, bi = hm + n + 1, a bi if a ⊕ b = 1,

(a)

where hm, ai and hn, bi are good sequences in A, (c) hGA , +, oi is the group defined by Chang. Proof. (a) Let {ak | k ∈ Z} be a good sequence in A and k ∈ Z. Because ak ⊕ak+1 = ak we get ak ∗ ∧ ak+1 = ak ∗ (ak ⊕ ak+1 ) = 0. Since A is linearly ordered, it follows that ak ∗ = 0 or ak+1 = 0. Thus ak = 1 or ak+1 = 0 for every k ∈ Z. Let m ∈ Z be the greatest element k ∈ Z such that ak = 1, which exists by Definition 5.1.2, condition (2). We get ak = 1 if k ≤ m and ak = 0 if k > m + 1. We proved that {ak | k ∈ Z} = hm + 1, am+1 i. (b) We have b∗ ≤ a or a ≤ b∗ , thus a ⊕ b = 1 or a b = 0. The desired result follows by Lemma 5.1.5 (1). (c) By (a), GA is the set of all the ordered pairs hm, ai with m ∈ Z and a ∈ A. By (b) and Lemma 5.1.5 (2), the group operations on GA coincide with Chang’s operations. The condition hm + 1, 0i = hm, 1i is obvious by Notation 5.1.4.

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63

Now we are able to prove that, for any MV-algebra A, the set of all the good sequences in A has a group structure. PROPOSITION 5.1.7. (a) The group operations from Definition 5.1.3 are well defined, i.e. the sum of good sequences is a good sequence and the opposite of a good sequence is a good sequence. (b)

hGA , +, oi is a group.

Proof. (a) Let a and b be two good sequences in A and m, n ∈ N such that condition (2) from Definition 5.1.2 is satisfied, respectively, for a and b. Then the same condition is trivially satisfied for the −a and n. The condition is also satisfied for a + b and m+n+1. This can be proved similarly with Lemma 5.1.5 (1), Cases 1 and 2. In order to prove that a + b and −a verify condition (1) from Definition 5.1.2 we will use Chang’s representation theorem which states that every MV-algebra is a subdirect product of MV-chains. Thus, it suffices to show that the desired equality holds in MV-chains. This immediately follows for −a by Lemma 5.1.5 (2), and for a + b by Proposition 5.1.6 (b). (b) Straightforward by Chang’s representation theorem and Proposition 5.1.6. In the sequel we will prove that GA is an `u-group. Let a and b be two good sequences in A. We define a ≤ b iff ak ≤ bk for all k ∈ Z. It is easy to see that ≤ is an order relation on GA . Moreover, since the MV-algebra operations are isotone, it follows that the group translation is isotone, so GA is a partially ordered group. We only have to define the lattice operations on GA . LEMMA 5.1.8. The lattice operations on GA are defined on components: (a ∨ b)k := ak ∨ bk

and

(a ∧ b)k := ak ∧ bk

for all k ∈ Z.

Proof. Firstly we will prove that the sequence a ∧ b is good. The condition (2) from Definition 5.1.2 is obviously satisfied. For every k ∈ Z we have (ak ∧ bk ) ⊕ (ak+1 ∧ bk+1 ) = (ak ⊕ ak+1 )∧(bk ⊕ bk+1 )∧(ak ⊕ bk+1 )∧(bk ⊕ ak+1 ) = ak ∧ bk ∧ (ak ⊕ bk+1 ) ∧ (bk ⊕ ak+1 ) = ak ∧ bk , so the condition (1) from Definition 5.1.2 holds. Moreover, a ∧ b is the infimum of a and b, since the order relation is defined on components. Since GA is a partially ordered group, the supremum of a and b is the good sequence given by a ∨ b = −(−a ∧ −b). For any k ∈ Z we get ∗

∗

(a ∨ b)k = (−(−a ∧ −b))k = ((−a ∧ −b)−k−1 ) = ((−a)−k−1 ∧ (−b)−k−1 ) ∗

= (ak ∗ ∧ bk ∗ ) = ak ∨ bk .



LEMMA 5.1.9. The good sequence u = h0, 1i is a strong unit of GA . Proof. By Lemma 5.1.5 (1), we get nu = hn − 1, 1i for every natural number n > 0. Thus, if a is a good sequence and n ∈ N such that ak = 0 for all k ≥ n and ak = 1 for all k < −n, it follows that a ≤ nu.

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We proved that GA is an `u-group for any MV-algebra A. Now we define the functor Ξ : MV → ALGu as follows: Ξ(A) := GA , Ξ(h) := e h where h : A → B is an MV-algebra homomorphism and e h({ak | k ∈ Z}) = {h(ak ) | k ∈ Z}. PROPOSITION 5.1.10. The MV-algebras A and [o, u]GA are isomorphic and the isomorphism is θA : A → [o, u]GA , defined by θA (a) = h0, ai for any a ∈ A. Proof. Let a be a good sequence in A such that o = h0, 0i ≤ a ≤ h0, 1i = u. Since the order relation is defined on components it follows that a = h0, ai with a ∈ A. Thus θA is well defined and bijective. Since θA (0) = h0, 0i = o we only have to prove that ∗ θA (a⊕b) = θA (a)⊕θA (b) and θA (a∗ ) = θA (a) for any a, b ∈ A. By Lemma 5.1.5 we ∗ obtain θA (a) = u − θA (a) = h0, 1i − h0, ai = h0, 1i + h−1, a∗ i = h0, a∗ i =θA (a∗ ) and also (θA (a) ⊕ θA (b))k = ((h0, ai + h0, bi) ∧ h0, 1i)k  1 if k < 0,    (a ⊕ b) ∧ 1 if k = 0, = (a b) ∧ 0 if k = 1,    0 if k > 1, = h0, a ⊕ bi = θA (a ⊕ b), Thus, θA is an MV-algebra isomorphism. In the sequel G will designate an Abelian `u-group with strong unit u, and A will designate [0, u]G . In order to complete the proof of the categorical equivalence between MV-algebras and `u-groups we have to show that G and GA are isomorphic `u-groups. DEFINITION 5.1.11. For any integer k let πk : G → A be defined by the rule πk (g) ≡ (g ∧ (k + 1)u) ∨ ku − ku = ((g − ku) ∧ u) ∨ 0. PROPOSITION 5.1.12. For every f, g ∈ G and k ∈ Z the following properties hold: (1)

π0 is the identity map on A,

(2)

if g ≤ ku, then πk (g) = 0,

(3)

if g ≥ (k + 1)u, then πk (g) = u,

(4)

if g ≤ 0, then πk (g) = 0 for all k ≥ 0,

(5)

if g ≥ 0, then πk (g) = u for all k < 0,

(6)

there must be some n ∈ N such that πk (g) = 0 for all k ≥ n and πk (g) = u for all k < −n,

(7)

πk (g) ≥ πk+1 (g) for all k ∈ Z,

(8)

πk (f ∨ g) = πk (f ) ∨ πk (g) and πk (f ∧ g) = πk (f ) ∧ πk (g) for all k ∈ Z,

(9)

πk (g) ⊕ πk+1 (g) = πk (g) for all k ∈ Z,

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65

∗

(10) πk (−g) = π−k−1 (g) for all k ∈ Z, L (11) πk (f + g) = i+j=k−1 (πi (f ) πj (g)) for all k ∈ Z, P (12) if g ≥ 0, then g = 0≤k πk (g), (13) πk (g+ ) = πk (g) for k ≥ 0 and πk (g+ ) = u for k < 0, ∗

(14) πk (g− ) = π−k−1 (g) for k ≥ 0 and πk (g− ) = u for k < 0, P P ∗ (15) g = 0≤k πk (g) − k<0 πk (g) , P (16) g = −n≤k
if all the necessary suprema and infima exist, then _  _ πk {gi | i ∈ I} = {πk (gi ) | i ∈ I} ^  ^ πk {gi | i ∈ I} = {πk (gi ) | i ∈ I}

Proof. (1), (2), (3) Follow directly from the definition of πk . (4) Follows from (2) (5) Follows from (3). (6) Since u is a strong unit for G, there is n ∈ N such that |g| ≤ nu, so −nu ≤ −|g| ≤ g ≤ |g| ≤ nu. The desired properties for πk follow from (2) and (3). (7) and (8) Follows directly from the definition of πk . (9) Denoting t := g − ku we get πk (g) ⊕ πk+1 (g) = [((t ∧ u) ∨ 0) + (((t − u) ∧ u) ∨ 0)] ∧ u = [((t∧) + ((t − u) ∧ u)) ∨ (t ∧ u) ∨ ((t − u) ∧ u) ∨ 0] ∧ u = [((2t − u) ∧ (t + u) ∧ t ∧ 2u) ∨ (t ∧ u) ∨ 0] ∧ u ≤ [(t ∧ 2u) ∨ (t ∧ u) ∨ 0] ∧ u = (t ∧ 2u ∧ u) ∨ (t ∧ u) ∨ 0 = (t ∧ u) ∨ 0 = πk (g) Since πk (g) ⊕ πk+1 (g) ≥ πk (g), the desired equality is proved. (10) Straightforward: ∗

π−k−1 (g) = u − [((g + (k + 1)u) ∧ u) ∨ 0] = [(u − (g + (k + 1)u)) ∨ 0] ∧ u = ((−g − ku) ∨ 0) ∧ u = ((−g − ku) ∧ u) ∨ 0 = πk (−g). (11) Will be (together with (12)) proved in R, since an equation holds in all the Abelian `-groups iff it holds in R. Given real numbers f and g, let i0 and j 0 be the unique integers such that i0 < f ≤ i0 + 1 and j 0 < g ≤ j 0 + 1. Then i0 + j 0 < f + g ≤ i0 + j 0 + 2,

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and πi (f ) πj (g) is 0 for i > i0 because πi (f ) = 0 andL for j > j 0 because πj (g) = 0. 0 0 Therefore if k > i + j + 1, then all terms in the sum i+j=k−1 (πi (f ) πj (g)) are 0 and so the sum itself is 0, which is in agreement with πk (f + g) because f + g ≤ i0 + f 0 + 2 ≤ k. If k ≤ i0 + j 0 − 1, then the sum contains the term πi0 −1 (f ) πk−i0 (g), and this term is 1 because each factor is 1. The result is that the entire sum is 1, which is in agreement with πk (f + g) because f + g > i0 + j 0 ≥ k + 1. Only two cases remain. If k = i0 + j 0 , then the sum reduces to (πi0 (f ) πj 0 −1 (g)) ⊕ (πi0 −1 (f ) πj 0 (g)) = πi0 (f ) ⊕ πj 0 (g) = ((f − i0 ) + (g − j 0 ))+ ∧ 1 = ((f + g) − k)+ ∧ 1 = πk (f + g). If k = i0 + j 0 + 1, then the sum reduces to πi0 (f ) πj 0 (g) = ((f − i0 ) + (g − j 0 ) − 1)+ = ((f + g) − k)+ ∧ 1 = πk (f + g). This completes the proof. (12) If g = 0, then the equality is obvious. Let g > 0 be a real number and n the first natural number such that n < g ≤ n + 1. By (2), πk (g) = 0 for k ≥ n + 1 so we have to prove that X g= πk (g). 0≤k≤n

We will prove this identity by induction on n. If n = 0, then g ∈ A and π0 (g) = g. Suppose n ≥ 1 and n + 1 < g ≤ n + 2. It follows that n < g − 1 < n + 1 so, by induction X g−1= πk (g − 1). 0≤k≤n

An easy computation shows that πk (g − 1) = πk+1 (g) and π0 (g) = (g ∧ 1) ∨ 0 = 1 ∨ 0 = 1 so X X g =1+ πk+1 (g) = πk (g). 0≤k≤n

(13)

Since g+ ≥ 0 we get πk (g+ ) = u for k < 0. If k ≥ 0 then = [((g ∨ 0) − ku) ∧ u] ∨ 0 = ((g − ku) ∧ u) ∨ ((−ku) ∧ u) ∨ 0 = ((g − ku) ∧ u) ∨ (−ku) ∨ 0 = ((g − ku) ∧ u) ∨ 0 = πk (g).

πk (g+ )

(14)

0≤k≤n+1

If k ≥ 0, then ∗

πk (g− )

= = = = = =

u − [(((−g) ∨ 0) − ku) ∧ u] ∨ 0 u − [(−g − ku) ∧ u] ∨ [(−ku) ∧ u] ∨ 0 u − [(−g − ku) ∧ u] ∨ (−ku) ∨ 0 u − [(−g − ku) ∧ u] ∨ 0 ((g + (k + 1)u) ∨ 0) ∧ u ((g + (k + 1)u) ∧ u) ∨ 0 = π−k−1 (g).

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67

(15) For any g ∈ G, g = g+ − g− . In the sequel we will use (12) for g+ and g− and the desired equality will follow by (14) and (15): g

= = = =

gP + − g− P π (g ) − π (g ) P0≤k k + P 0≤k k − ∗ π (g) − π (g) P0≤k k P0≤k −k−1∗ π (g) − π (g) . 0≤k k k<0 k

(16) Let n ∈ N such that πk (g) = 0 for all k ≥ n and πk (g) = u for all k < −n. By (15) we get: P P ∗ g = π (g) − k<0 πk (g) P0≤k k P ∗ = π (g) − −n≤k<0 πk (g) P0≤k
We will prove the identity for suprema. W W W πk ( {gi | i ∈ I}) = ((( W gi ) − ku) ∧ u) ∨ 0 = W (( (gi − ku)) ∧ u) ∨ 0 = ((gi − ku) ∧ u ∨ 0) = {πk (gi ) | i ∈ I}.

The identity for infima follows similarly. The crucial point is that each g ∈ G can be uniquely coded as the sequence {πk (g) | k ∈ Z} in A. LEMMA 5.1.13. For any element g ∈ G the sequence {πk (g) | k ∈ Z} is the unique good sequence {ak | k ∈ Z} in A such that X X g= ak − ak ∗. 0≤k

k<0

Proof. The fact that {πk (g) | k ∈ Z} is a good sequence for g which satisfies the intended equality follows from Proposition 5.1.12 (9), (6) and (15). Suppose that {ak } and {bk } are good sequences for g such that X X X X g= ak − ak ∗ = bk − bk ∗. 0≤k

k<0

0≤k

k<0

We can assume that there is some n ∈ N such that ak = bk = 0 for all k ≥ n and ak = bk = u for all k < −n, so X X g= ak − nu = bk − nu. −n≤k
−n≤k
We get a−n =

M −n≤k
=

X −n≤k
ak =

X

ak ∧ u

−n≤k
bk ∧ u =

M

bk = b−n .

−n≤k
By induction it follows that ak = bk for any −n ≤ k < n.

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PROPOSITION 5.1.14. The `u-groups hG, ui and hGA , ui are isomorphic and the isomorphism is ηG : G → GA , defined by ηG (g) = {πk (g) | k ∈ Z}. Proof. It is obvious that ηG (0) = o and ηG (u) = u. Moreover, from Definition 5.1.3 and Proposition 5.1.12 (10) and (11), it follows that ηG (−g) = −ηG (g) and ηG (g+h) = ηG (g)+ηG (h), so ηG is an `u-group homomorphism. If g and h ∈ G such that ηG (g) = ηG (h), then πk (g) = πk (h) for every k ∈ Z so, by Proposition 5.1.12 (15),P g = h and ηP G is injective. Let a = {ak | k ∈ Z} be a good sequence in GA and g = 0≤k ak − ∗ k<0 ak . By Lemma 5.1.13, ak = πk (g) for every k ∈ Z, so ηG (g) = {ak | k ∈ Z}. Thus ηG is also surjective. We prove that ηG is an `u-group isomorphism between hG, ui and hGA , ui. COROLLARY 5.1.15. Γ(ηG ) = θΓ(G,u) . Proof. The proof is straightforward using the definitions. The proof of the categorical equivalence between Abelian `-groups with strong unit and MV-algebras is finished by the following theorem. THEOREM 5.1.16. The functors Γ : ALGu → MV and Ξ : MV → ALGu establish a categorical equivalence. Proof. In Proposition 5.1.10 we proved that, for any MV-algebra A, there is an MValgebras isomorphism, θA , between A and Γ(Ξ(A)). By Proposition 5.1.14, for any `u-group hG, ui there is an `u-group isomorphism, ηG , between G and Ξ(Γ(G, u)). Let A and B be two MV-algebras and h : A → B an MV-algebra homomorphism. We have to prove that Γ(Ξ(h)) ◦ θA = θB ◦ h. For a ∈ A we get Γ(Ξ(h)) ◦ θA (a) = Ξ(h)(h0, ai) = e h(h0, ai) = h0, h(a)i = θB (h(a)). Similarly, if hG, ui and hH, vi are two `u-groups and h : hG, ui → hH, vi is an `ugroup homomorphism, then we will show that Ξ(Γ(h)) ◦ ηG = ηH ◦ h. Since h is a homomorphism, we have h(πk (g)) = πk (h(g)) for every g ∈ G and k ∈ Z. Thus, for any g ∈ G, we get g Ξ(Γ(h)) ◦ ηG (g) = Ξ(Γ(h))({πk (g) | k ∈ Z}) = Γ(h)({π k (g) | k ∈ Z}) = {Γ(h)(πk (g)) | k ∈ Z} = {h(πk (g)) | k ∈ Z} = {πk (h(g)) | k ∈ Z} = ηh (h(g)). The proof of the categorical equivalence is now complete. 5.2

Properties of Γ(G, u)

In the sequel hG, ui is an `u-group and A = Γ(G, u). We recall that an `u-subgroup of G is a subset H ⊆ G which contains u and it is closed under the `-group operations. PROPOSITION 5.2.1. (a)

If H ⊆ G is an `u-subgroup of G, then H ∩ A is an MV-subalgebra of A. Moreover, H is the `u-subgroup generated by H ∩ A in G.

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(b)

69

If B ⊆ A is an MV-subalgebra of A , then H = {g ∈ G | πk (g) ∈ B for all k ∈ Z} is the `u-subgroup generated by B in G. Moreover, H ∩ A = B.

(c)

There is a bijective correspondence between the set of all the MV-subalgebras of A and the set of all the `u-subgroups of G.

Proof. (a) Let H be an `u-subgroup of G. It is easy to see that H ∩ A is an MVsubalgebra of A. In order to prove that H is the `u-subgroup generated by H ∩ A in G, we consider V another `u-subgroup of G that contains H ∩ A and h ∈ H. Since H is an `-subgroup which contains u, it follows that πk (h) ∈ H for any k ∈ Z, so πk (h) ∈ H ∩ A ⊆ V for any k ∈ Z. By Proposition 5.1.12 (16), we get h ∈ V . Thus H ⊆ V and H is the smallest `u-subgroup that contains H ∩ A. (b) Since B is closed under the MV-algebras operations, by Proposition 5.1.12 (10), (11) and (17), we infer that H is an `-subgroup of G. By Proposition 5.1.12 (1), we get B ∈ H, so u ∈ H. Thus, H is an `u-subgroup of G that includes B. Let V be another `u-subgroup of G containing B and h ∈ H. We have πk (h) ∈ B ⊆ V for any k ∈ Z, so h ∈ V by Proposition 5.1.12 (16). Hence H ∈ V and we proved that H is the `u-subgroup generated by B in G. Let h be an element in H ∩ A. Because h ∈ A we get πk (h) = h for any k ∈ Z. Since h ∈ H, we get πk (h) ∈ B for any k ∈ Z. Thus h ∈ B, so H ∩ A = B. (c) Straightforward by (a) and (b). PROPOSITION 5.2.2. (a)

If H ⊆ G is an `-ideal of G, then H ∩ A is an ideal of A. Moreover, H = (H ∩ A]G , where (H ∩ A]G is the `-ideal generated by H ∩ A in G.

(b)

If I ⊆ A is an ideal of A, then (I ]G = {g ∈ G | πk (|g|) ∈ I for all k ≥ 0} = {g ∈ G | |g| ∧ u ∈ I} is the `-ideal generated by I in G. Moreover, (I ]G ∩ A = I.

(c)

If I, J ⊆ A are two ideals, then I⊆J

(d)

iff (I ]G ⊆ (J ]G .

There is a bijective correspondence between the set of all the ideals of A and the set of all the `-ideals of G, which maps the maximal ideals of A into maximal ideals of G.

Proof. Let I be an ideal of A. Note that π0 (|g|) = |g| ∧ u for any g ∈ G. By Proposition 5.1.12 (7), it follows that |g| ∧ u ∈ I iff πk (|g|) ∈ I for all k ≥ 0. In the sequel we will only prove that H = {g ∈ G | πk (|g|) ∈ I for all k ≥ 0}

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is an `-ideal of G. We recall that, in an `-group G, |g| = g ∨ (−g), |g + h| ≤ |g| + |h| and |g ∨ h| ≤ |g| ∨ |h| ≤ |g| + |h| for any g, h ∈ G. Since |g| = | − g| for any g ∈ G we have g ∈ H implies −g ∈ H. For g, h ∈ H and k ≥ 0 we get πk (|g| + |h|) ∈ I by Proposition 5.1.12 (11) and the fact that I is an ideal. Since πk (|g + h|) ≤ πk (|g| + |h|), it follows that πk (|g + h|) ∈ I for any k ≥ 0, so g + h ∈ H. Similarly, we can prove that g ∨ h ∈ H, since πk (|g ∨ h|) ≤ πk (|g| + |h|) ∈ I for any k ≥ 0. Thus, H is an `-subgroup of G. In order to prove that H is convex, we consider 0 ≤ g ≤ h ∈ H. It follows that |g| = g, |h| = h and πk (|g|) ≤ πk (|h|) ∈ I for any k ≥ 0. Since I is an ideal, we get πk (|g|) ∈ I for any k ≥ 0, so g ∈ H. We proved that H is an `-ideal of G. The rest of the proof is straightforward. COROLLARY 5.2.3. The topological spaces Max (A) and Max (G) (with the spectral topologies) are homeomorphic. Proof. It follows directly by Proposition 5.2.2 (d). PROPOSITION 5.2.4. Let hG, ui be an `u-group. The MV-algebra A is an MV-chain iff G is linearly ordered. Proof. If G is linearly ordered, then A is linearly ordered, since the order relation on A is a restriction of the order relation on G. Now, suppose that A is linearly ordered. By Proposition 5.1.6 and Proposition 5.1.12 (16) we infer that for every g ∈ G there is a ∈ A and m ∈ Z such that g = a + mu. Moreover, if g and v are in G, g = a + mu and v = b + nu where a, b ∈ A and m, n ∈ Z, then g≤v

iff m ≤ n or m = n and a ≤ b.

This is obviously a linear order relation. If a is an element of A we will denote (na)A = a ⊕ · · · ⊕ a | {z } n

and

(na)G = a + · · · + a . | {z } n

The two values can be different: for example (nu)A = u < (nu)G for n ≥ 2. Still, if there is no possible confusion or the two values coincide, we will not use any determination. In the sequel, we define the divisible MV-algebras, which will be essentially used in the proof of Chang’s Completeness Theorem for MV-algebras (Theorem 5.3.7). PROPOSITION 5.2.5. For an MV-algebra A the following are equivalent: (a)

for any a ∈ A and for any natural number n ≥ 1 there is x ∈ A such that nx = a and a∗ ⊕ (n − 1)x = x∗ ,

(b)

for any a ∈ A and for any natural number n ≥ 1 there is x ∈ A such that (nx)+ is defined and (nx)+ = nx = a, where + is the partial addition defined in Section 2.9.

Proof. We have x∗ = a∗ ⊕(n−1)x = (nx)∗ ⊕(n−1)x = (x⊕(n−1)x)∗ ⊕(n−1)x = x∗ ((n − 1)x)∗ ⊕ (n − 1)x = x∗ ∨ (n − 1)x. Thus, the condition a∗ ⊕ (n − 1)x = x∗ is equivalent with (n − 1)x ≤ x∗ .

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(a) ⇒ (b) According to the definition of the partial addition (n − 1)x + x is defined and (n − 1)x + x = (n − 1)x ⊕ x = nx. Moreover, for any 1 ≤ k ≤ n − 1 we have kx ≤ (n − 1)x ≤ x∗ , so kx + x is defined and kx + x = kx ⊕ x = (k + 1)x. Thus, a = nx = (n − 1)x + x = (n − 2)x + x + x = · · · = (nx)+ . (b) ⇒ (a) Since (nx)+ is defined, we get ((n − 1)x)+ is defined and ((n − 1)x)+ = (n − 1)x ≤ x∗ . The desired conclusion is straightforward. DEFINITION 5.2.6. An MV-algebra A is called divisible if it satisfies one of the equivalent condition from Proposition 5.2.5. EXAMPLE 5.2.7. The MV-algebras [0, 1], Z ∩ [0, 1] and Łn+1 are divisible. PROPOSITION 5.2.8. The MV-algebra A is divisible iff G is a divisible `-group. Proof. Note that the partial addition on A defined in Section 2.9 coincide with the group addition. Let G be a divisible `-group, a an arbitrary element from A and n ≥ 1. Then there is x ∈ G such that (nx)G = a. We will prove that x ≥ 0. In G we have n(x ∧ 0)G = (nx)G ∧ ((n − 1)x)G ∧ · · · ∧ 0 = a ∧ ((n − 1)x)G ∧ · · · ∧ 0 = ((n − 1)x)G ∧ · · · ∧ 0 = (n − 1)(x ∧ 0)G . It follows that x ∧ 0 = 0, so x ≥ 0. Thus x ≤ nx ≤ a, so x ∈ A and (nx)G = (nx)A = a. Conversely, let A be a divisible P MV-algebra and g ∈ G. By Proposition 5.1.12 (16) there is n ∈ N such that g = −n≤l
An MV-algebra A is semisimple iff G is an Archimedean

Proof. Suppose G is Archimedean and let a ∈ A such that (na)A ≤ a∗ for any n ∈ N . If a∗ = 1, then a = 0. If a∗ < 1, then (na)A = (na)G ≤ a∗ . By hypothesis it follows that a ≤ 0. Since a is also positive we get a = 0, so A is Archimedean. Conversely, suppose that A is semisimple, hence A is an Archimedean MV-algebra. Let g, v ∈ G such that g, v ≥ 0 and ng ≤ v for any n ∈ N . We know that there are k ∈ N and a1 , . . . , ak in A such that v = a1 + · · · + ak . We will prove that g = 0 by induction on k. If k = 1, then v, g ∈ A so (ng)G = (ng)A ≤ v ≤ u. For any n ≥ 1 we get (n − 1)g ≤ u − g = g ∗ so, by hypothesis, we get g = 0. In order to prove the induction step we suppose that our assertion holds for any g, v ≥ 0 such that v is a sum of less than k elements from A and we assume that ng ≤ v = a1 + · · · + ak−1 + a for any n ∈ N , where a1 , . . . , ak−1 , a ∈ A. For n ∈ N we get ng − a ≤ a1 + · · · + ak−1 . If we denote t = ng − a, then, for any m ∈ N , we have mt = mng − ma ≤ mng ≤ a1 + · · · + ak−1 . Thus m(t ∨ 0) = mt ∨ (m − 1)t ∨ · · · ∨ t ∨ 0 ≤ a1 + · · · + ak−1 for any m ∈ N . By induction hypothesis, t ∨ 0 = 0 so t ≤ 0, i.e. ng ≤ a. Since n was arbitrary in N , it follows that ng ≤ a for any n ∈ N . Using the induction hypothesis for k = 1 we conclude that g = 0.

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5.3

Chang’s completeness theorem

Following Theorem 5.1.16, we can identify an MV-algebra A with the interval [0, u] of some `u-group. COROLLARY 5.3.1. Any MV-algebra can be embedded into a divisible MV-algebra. In particular, any MV-chain can be embedded into a divisible MV-chain. Proof. Let A be an MV-algebra and hG, ui an `u-group such that A is isomorphic to [0, u]G . By [3, Chapter 3], there is an Abelian divisible `-group H and an `-group embedding h : G → H. Moreover, if G is totally ordered, then H is totally ordered by [34]. Since h is injective, we get h(u) > 0, so we can consider the MV-algebra D = [0, h(u)]H . It is obvious that the restriction of h to [0, u] ⊆ G is an MV-algebra embedding of [0, u]G into D. Thus, the MV-algebra A can be embedded into D. By Proposition 5.2.8, D is a divisible MV-algebra. If A is an MV-chain, then D will be also an MV-chain by Proposition 5.2.4. Recall now the first-order theories of MV-algebras and `-groups. FACT 5.3.2 (The theory of MV-algebras). The language of MV-algebras is ΣMV = {0, ¬, ⊕} where 0 is a constant symbol, ¬ is a unary function symbol and ⊕ is a binary function symbol. The theory of MV-algebras has the following axioms: (1) (2) (3) (4) (5) (6)

(∀xyz)x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z, (∀x)x ⊕ 0 = x, (∀xy)x ⊕ y = y ⊕ x, (∀x)¬(¬x) = x, (∀x)x ⊕ (¬0) = ¬0, (∀xy)x ⊕ ¬(x ⊕ ¬y) = y ⊕ ¬(y ⊕ ¬x).

If we add the axiom (7)

(∀xy)(x ⊕ ¬(x ⊕ ¬y) = x ∨ x ⊕ ¬(x ⊕ ¬y) = y)

we get the theory of MV-chains. Moreover, the theory of divisible MV-chains is obtained adding an infinite number of axioms: for any n ≥ 1 we add the axiom (8n ) (∀x)(∃y)(ny = x ∧ (¬x) ⊕ (n − 1)y = ¬y). Note that the interpretation of ¬ in an MV-algebra is the unary operation ∗ . FACT 5.3.3 (The theory of Abelian `-groups). The language of `-groups is Σlg = {0, −, +, ∨, ∧, ≤} where 0 is a constant symbol, − is a unary operation, +, ∨, ∧ are binary function symbols and ≤ is a binary relation symbol. The theory of Abelian `-groups has the following axioms: (1) (2) (3) (4)

(∀xyz)x + (y + z) = (x + y) + z, (∀x)x + 0 = x, (∀x)x + (−x) = 0, (∀xy)x + y = y + x,

Chapter I: Łukasiewicz Logic and MV-Algebras

(5) (6) (7) (8) (9) (10)

73

(∀x)x ≤ x, (∀xy)(x ≤ y ∧ y ≤ x → x = y), (∀xyz)(x ≤ y ∧ y ≤ z → x ≤ z), (∀xy)(x ∨ y = y ↔ x ≤ y), (∀xy)(x ∧ y = x ↔ x ≤ y), (∀xyz)(x ≤ y → (x + z) ≤ (y + z)).

It is possible to give other descriptions using either the relation symbol ≤ or the function symbols ∨ and ∧. Moreover, we can remove − from our language and replace the axiom (3) with axiom (30 ) (∀x)(∃y)x + y = 0. The axioms we proposed are more convenient for our purpose. The theory of totally ordered Abelian `-groups is obtained adding the axiom (11) (∀xy)(x ≤ y ∨ y ≤ x). The theory of totally ordered divisible Abelian `-groups can be described adding for any n ≥ 1 the axiom (12n ) (∀x)(∃y)(ny = x). The following result is a generalization of [13, Lemma 7]. PROPOSITION 5.3.4. For any sentence σ of ΣMV there is a formula with only one free variable σ e(v) of Σlg such that for any MV-algebra A we have A |= σ

iff G |= σ e[u]

for any Abelian `-group G and u > 0 in G such that A is isomorphic with [0, u]G . Proof. Since any two isomorphic MV-algebras are elementarily equivalent, it suffices to prove the desired result for A = [0, u]G , where G is an Abelian `-group and u > 0 in G. Let t(v1 , . . . , vk ) be a term of ΣMV and v a propositional variable different from v1 , . . ., vk . We define e t as follows: – if t = 0, then e 0 is 0, – if t = ¬t1 , then e t is v − te1 , – if t = t1 ⊕ t2 , then e t is (t1 + t2 ) ∧ v. Let ϕ(v1 , . . . , vk ) be a formula of ΣMV such that all the free and bound variables of ϕ are in {v1 , . . . , vk } and v a propositional variable different from v1 , . . ., vk . We define ϕ e as follows: e = t2, e – if ϕ is t1 = t2 , then ϕ e is t1 e – if ϕ is ¬ψ, then ϕ e is ¬ψ, – if ϕ is ψ ∨ χ, then ϕ e is ψe ∨ χ e and similarly for ∧, →, ↔, e – if ϕ is (∀vi )ψ, then ϕ e is (∀vi )(0 ≤ vi ∧ vi ≤ v → ψ), e – if ϕ is (∃vi )ψ, then ϕ e is (∃vi )(0 ≤ vi ∧ vi ≤ v → ψ). Thus to any formula ϕ(v1 , . . . , vk ) of ΣMV corresponds a formula ϕ(v e 1 , . . . , vk , v) of Σlg . As a consequence, to any sentence σ of ΣMV corresponds a formula with only one free variable σ e(v) of Σlg .

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Since A = [0, u]G , we recall that, for any a, b ∈ A we have a ⊕ b = (a + b) ∧ u and a∗ = u − a. We prove that for any term t(v1 , . . . , vn ) of ΣMV and any a1 , . . . , an ∈ A, t[a1 , . . . , an ] = e t[a1 , . . . , an , u]. Thus, – –

if t = 0 it is obvious, if t = ¬t1 , then e t[a1 , . . . , an , u] = u − te1 [a1 , . . . , an , u] = u − t1 [a1 , . . . , an ] = t1 [a1 , . . . , an ]∗ = (¬t1 )[a1 , . . . , an ] = t[a1 , . . . , an ],

–

if t = t1 ⊕ t2 , then e t[a1 , . . . , an , u] =

(te1 [a1 , . . . , an , u] + te2 [a1 , . . . , an , u]) u(t1 [a1 , . . . , an ] + t2 [a1 , . . . , an ]) ∧ u = (t1 [a1 , . . . , an ] ⊕ t2 [a1 , . . . , an ]) = t[a1 , . . . , an ].

Now, we will prove that for any formula ϕ(v1 , . . . , vn ) of ΣMV and any a1 , . . . , an ∈ A, A |= ϕ[a1 , . . . , an ] iff G |= ϕ[a e 1 , . . . , an , u]. If ϕ is of the form t1 = t2 or ¬ψ or ψ ∨ χ the proof is straightforward. We give the details when ϕ is (∀vi )ψ. The case when ϕ is (∃vi )ψ is similar, since a ∈ A is equivalent to 0 ≤ a ≤ u in G. Thus, if ϕ is (∀vi )ψ, then G |= ϕ[a e 1 , . . . , an , u]

iff iff iff iff iff

for any a ∈ G, 0 ≤ a ≤ u implies G |= ψ[a1 , . . . , ai−1 , a, ai+1 , . . . , an , u] for any a ∈ G, 0 ≤ a ≤ u implies A |= ψ[a1 , . . . , ai−1 , a, ai+1 , . . . , an ] for any a ∈ A, A |= ψ[a1 , . . . , ai−1 , a, ai+1 , . . . , an ] A |= (∀vi )ψ[a1 , . . . , an ] A |= ϕ[a1 , . . . , an ].

The desired conclusion is obtained in the particular case when ϕ is a sentence. The following result is known for Abelian `-groups. THEOREM 5.3.5 ( [79, Theorem 3.1.2]). Any non-trivial divisible totally ordered `group is elementarily equivalent with the group Q of rationals. We prove a similar result for MV-algebras. THEOREM 5.3.6. Any non-trivial divisible MV-chain is elementarily equivalent with [0, 1]Q . Proof. Let A be a divisible MV-chain and hG, ui an `u-group such that A is isomorphic with Γ(G, u) = [0, u]G . By Proposition 5.2.4 and 5.2.8, G is a totally ordered divisible `u-group. We have to prove that for any sentence σ of ΣMV , A |= σ iff [0, 1]Q |= σ. If A |= σ then, by Proposition 5.3.4, G |= σ e[u] where σ e(v) is a formula of Σlg with only one free variable. If τ is the formula (∃v)(0 ≤ v ∧ (¬(v = 0)) ∧ σ e) then τ is a sentence and G |= τ . By Theorem 5.3.5, Q |= τ , so there is a rational number r ∈ Q such that r > 0 and Q |= σ e[r]. Using again Proposition 5.3.4, the MV-algebra

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75

[0, r]Q |= σ. If we define f : [0, 1] → [0, r] by f (x) = x · r, where · is the real numbers product operation, then it is obvious that f is a bijective function. Moreover, f is an MV-algebra isomorphism between [0, 1]Q and [0, r]Q . It follows that [0, 1]Q and [0, r]Q are elementary equivalent, so [0, 1]Q |= σ. Conversely, let us suppose that [0, 1]Q |= σ. Using our previous observation, [0, 1]Q is isomorphic to [0, r]Q for any rational number r > 0. Thus, [0, r]Q |= σ for any r > 0. By Proposition 5.3.4, Q |= σ e[r] for any r > 0. If θ is the sentence (∀v)(0 ≤ v ∧ (¬(v = 0)) → σ e), then Q |= θ. By Theorem 5.3.5 we infer that G |= θ. Since u > 0 in G, it follows that G |= σ e[u] which is equivalent to A |= σ. In fact, the previous theorem asserts that the theory of nontrivial divisible MVchains is complete. We are ready to prove Chang’s completeness theorem. Note that the present proof uses the full equivalence between MV-algebras and `u-groups, as well as model-theoretical properties. In [18] one can see a self-contained proof, which involves only a minimal fragment of Γ-theory. THEOREM 5.3.7 (Chang’s completeness theorem). For any sentence σ of ΣMV , A |= σ for any MV-algebra A iff [0, 1]Q |= σ iff [0, 1] |= σ. Proof. One implication is obvious. To prove the other one, let A be an MV-algebra such that A 6|= σ. By Theorem 4.1.4 and Corollary 5.3.1, we can safely assume that A is a divisible MV-chain. Hence, using Theorem 5.3.6, we get that [0, 1]Q 6|= σ. The equivalence with [0, 1] |= σ follows by Theorem 5.3.6, since [0, 1] is a non-trivial divisible MV-chain. 5.4

Functional representation of semisimple MV-algebras

In this section we give concrete representations for finite, simple and semisimple MV-algebras. PROPOSITION 5.4.1. Any simple MV-algebra is isomorphic to a subalgebra of [0, 1]. Proof. By Propositions 4.2.9 and 4.2.5 it follows that any simple MV-algebra A is linearly ordered and Archimedean. Thus, following Propositions 5.2.4 and 5.2.9, we infer that the corresponding `u-group hG, ui is linearly ordered and Archimedean. By Holder’s Theorem, there is a subgroup H of real numbers such that G and H are isomorphic. If h ∈ H is the corresponding element for u, then h is a strong unit of H. Consider H 0 = {x/h | x ∈ H} where / stands for the real numbers division. One can easily see that H 0 is isomorphic to H and 1 is a strong unit for H 0 . We get A ' Γ(G, u) ' Γ(H, h) ' Γ(H 0 , 1). Thus, A is isomorphic to a subalgebra of [0, 1]. COROLLARY 5.4.2. Any finite and simple MV-algebra is isomorphic to Łn for some n ≥ 2. Proof. By Proposition 5.4.1 and Corollary 4.1.11. COROLLARY 5.4.3. Any finite MV-chain is isomorphic to Łn for some n ≥ 2. Proof. If A is a finite MV-chain, then, by Proposition 4.2.11 (b), A is a simple MValgebra. the desired result follows by Corollary 5.4.2.

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PROPOSITION 5.4.4. For any finite MV-algebra A the following properties hold: (a)

Spec(A) = Max (A),

(b)

A is semisimple,

(c) A ' Łk1 × · · · × Łkn for some natural numbers n ≥ 1 and k1 , . . . , kn ≥ 2. Proof. Let A be a finite MV-algebra. (a) For any prime ideal P ⊆ Spec(A), the quotient A/P is a finite MV-chain. By Corollary 5.4.3, A/P is a simple MV-algebra, so P is a maximal ideal. (b) Straightforward by (a). (c) We have M1 ∩ · · · ∩ Mn = {0}, where Max (A) = {M1 , . . . , Mn }. If i 6= j, then Mi ⊂ Mi ∨ Mj , so Mi ∨ Mj = A. Now we can apply Corollary 3.2.12 and we get that A is isomorphic to the direct product (A/M1 ) × · · · × (A/Mn ). For any i ∈ {1, . . . , n}, A/Mi is a finite simple MV-algebra. the desired result follows by Corollary 5.4.2. We focus now on the representation of semisimple MV-algebras. DEFINITION 5.4.5. For a set X, the MV-algebra [0, 1]X will be called the bold algebra of fuzzy subsets of X. Assume A is a semisimple MV-algebra. Then for any M ∈ Max (A), by Propositions 4.2.10 and 5.4.1, A/M is isomorphic with a subalgebra of [0, 1]. Hence for any a ∈ A the congruence class [a]M is mapped into an element a ˆM ∈ [0, 1]. In this, for any a ∈ A, we define a map a ˆ : Max (A) → [0, 1], a ˆ(M ) = a ˆM for any M ∈ Max (A). THEOREM 5.4.6. For a semisimple MV-algebra A, the function ιA : A → [0, 1]Max (A) defined as ιA (a) = a ˆ for any a ∈ A is an MV-algebra embedding. As consequence, any semisimple MV-algebra is isomorphic with a subalgebra of a bold algebra of fuzzy sets. Proof. It is straightforward by Propositions 3.2.11, Proposition 5.4.1 and the above considerations. By Theorem 3.6.10, the maximal ideal space Max (A) endowed with the spectral topology is a compact Hausdorff space. The MV-algebra of continuous functions C(Max (A), [0, 1]) with pointwise operations is an MV-subalgebra of [0, 1]Max (A) . We prove that, for any a ∈ A, the function a ˆ : Max (A) → [0, 1] is continuous. As consequence, we represent the semisimple MV-algebra A an MV-algebra of [0, 1]-valued continuous functions defined on its maximal ideal space. THEOREM 5.4.7. Any semisimple MV-algebra A is, up to isomorphism, an MV-algebra of [0, 1]-valued continuous functions defined on some nonempty compact Hausdorff space X (with pointwise operations). Moreover, A is separating, i.e. for any x 6= y in X there is f ∈ A such that f (x) 6= f (y).

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Proof. Using the notations from Theorem 5.4.6, we firstly prove that ιA (a) = a ˆ is continuous for any a ∈ A. Without loss of generality one can assume that A = Γ(G, u) for some `u-group hG, ui. For a ∈ A ⊆ G one can easily prove that [a]M = [a](M ]G for any maximal ideal M ∈ Max (A). By Corollary 5.2.3 there exists a homeomorphism h : Max (G) → Max (A) and we define the function a ˜ : Max (G) → [0, 1], a ˜=a ˆ ◦ h. Since a ˜ is a continuous function by [7, Theorem 13.2.4], it follows that a ˆ is also a continuous function. Hence ιA (A) is an MV-subalgebra of C(Max (A), [0, 1]). In order to prove that ιA (A) is separating, assume that M 6= N in Max (A). There is a ∈ A such that a ∈ M and a ∈ / N , so a ˆ(M ) = 0 6= a ˆ(N ). Perfect MV-algebras and Abelian `-groups

5.5

We will prove that the category of perfect MV-algebras is equivalent with the category of Abelian `-groups. Let Perf denote the category whose objects are perfect MValgebras and whose morphisms are MV-algebra homomorphisms. Note that Perf is a full subcategory of MV. Denote ALG the category whose objects are Abelian `-groups and whose morphisms are `-group homomorphism. The above mentioned categorical equivalence will be established by the functors ∆ : ALG → Perf

and

D : Perf → ALG

defined as in the following. For an Abelian `-group G, let ∆(G) be the perfect MValgebra from Example 4.3.33. If h : G → H is an `-group homomorphism, we define ∆(h) : ∆(G) → ∆(H) and

∆(h)(k, g) := hk, h(g)i for any hk, gi ∈ ∆(G).

One can easily prove that ∆(h) is an MV-algebra homomorphism and ∆ is a functor. In order to define the functor D we need to remind some classical result of `-group theory. THEOREM 5.5.1. Let M be a partially ordered Abelian monoid. Then M is the positive cone of a partially ordered Abelian group iff the following properties hold: (a)

M is cancellative,

(b)

the order on M is natural, i.e. a ≤ b iff there exists x ∈ M such that b = a + x.

Proof. We only remind the construction of G. For a detailed proof see [34]. Define the following equivalence on M × M : ha, bi ∼ hc, di iff a + d = c + b and we consider G = M × M/∼, the set of all the equivalence classes. If we denote by [a, b] the equivalence class of the element ha, bi, then the group operations on G will be defined by: [a, b] + [c, d] = [a + c, d + b] and − [a, b] = [b, a]. It is obvious that M can be identified with the set {[a, 0] | a ∈ M }. The order on G is defined by [a, b] ≤ [c, d] iff [c, d] − [a, b] ∈ M and − [a, b] + [c, d] ∈ M. One can prove that hG, +, [0, 0], ≤i is an Abelian partially ordered group and M is isomorphic with the positive cone of G.

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FACT 5.5.2. Under the hypothesis of the previous theorem, if M is a lattice ordered monoid then G is an Abelian `-group. By [8], a partially ordered group G is an `-group iff for every g ∈ G, g+ = g ∨ 0 exists in G. Then g ∨ h = (g − h)+ + h and g ∧ h = −((−h) ∨ (−g)) for every g, h ∈ G. In our particular case one can prove that [a, b]+ = [a ∨ b, b]. It follows that the lattice operations on G are defined by [a, b] ∨ [c, d] = [(a + d) ∨ (c + b), d + b]

and [a, b] ∧ [c, d] = [a + c, (d + a) ∨ (b + c)]

If A is a perfect MV-algebra then, by Corollary 3.5.3, Rad (A) is a latticial monoid that satisfies the hypothesis of Theorem 5.5.1. Let D(A) := Rad (A) × Rad (A)/∼ be defined as in the proof of Theorem 5.5.1. Hence D(A) is an Abelian `-group such that D(A)+ and Rad (A) are isomorphic latticial monoids. If A and B are perfect MV-algebras and f : A → B is an MV-algebra homomorphism we define a mapping D(f ) : D(A) → D(B) as D(f )([a, b]) := [f (a), f (b)] for any [a, b] ∈ D(A). Hence D(f ) is a well-defined `-group homomorphism. One can easily prove that D is a functor. PROPOSITION 5.5.3. If A is a perfect MV-algebra, then A and ∆(D(A)) are isomorphic MV-algebras. Proof. Let us denote G = D(A) and B = ∆(G). It follows that Rad (A) and G+ are isomorphic lattice monoids. By Example 4.3.33, B is a perfect MV-algebra and G+ is also isomorphic with Rad (B). So there exists f : Rad (A) → Rad (B) an isomorphism of lattice monoids. We define F : A → B by  f (a) if a ∈ Rad (A), F (a) = ∗ ∗ f (a ) if a ∈ Rad (A)∗ . It is obvious that F is bijective. We only have to prove that F is an MV-algebra homomorphism. Note that F (0) = f (0) = 0 and F (1) = f (0)∗ = 1. Since b = b∗∗ for every b ∈ B, we get  f (a)∗ if a ∈ Rad (A), ∗ F (a) = ∗ ∗∗ f (a ) if a ∈ Rad (A)∗  f (a)∗ if a ∈ Rad (A), = f (a∗ ) if a ∈ Rad (A)∗  f (a)∗ if a∗ ∈ Rad (A)∗ , = f (a∗ ) if a∗ ∈ Rad (A) = F (a∗ ). We will prove that F (a ∨ b) = F (a) ∨ F (b). If a, b ∈ Rad (A), then the relation is obvious, since f is a lattice morphism. If a, b ∈ Rad (A)∗ , then F (a ∨ b) = f (a∗ ∧ b∗ )∗ = (f (a∗ ) ∧ f (b∗ ))∗ = f (a∗ )∗ ∨ f (b∗ )∗ = F (a) ∨ F (b). If a ∈ Rad (A) and b ∈ Rad (A)∗ , then F (a) ∈ Rad (B) and F (b) ∈ Rad (B)∗ . By Lemma 3.5.2, a ∨ b = b and F (a) ∨ F (b) = F (b), so F (a ∨ b) = F (a) ∨ F (b).

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Now we are able to prove that F (a ⊕ b) = F (a) ⊕ F (b). If a, b ∈ Rad (A) the equality is obvious since f is a homomorphism of monoids. If a, b ∈ Rad (A)∗ , then F (a), F (b) ∈ Rad (B)∗ . By Lemma 3.5.2 (a), a ⊕ b = 1 and F (a) ⊕ F (b) = 1 so the desired relation is obvious. If a ∈ Rad (A) and b ∈ Rad (A)∗ , then we get: F (a ∨ b∗ ) = F (a) ∨ F (b∗ ), F ((b∗ a∗ ) ⊕ a) = (F (b∗ ) F (a)∗ ) ⊕ F (a), F (b∗ a∗ ) ⊕ F (a) = (F (b∗ ) F (a∗ )) ⊕ F (a). To prove the last equality we use the fact that b∗ a∗, a ∈ Rad (A). The last equality is in Rad (B), which is a cancellative monoid, so F (b∗ a∗ ) = F (b∗ ) F (a∗ ). Thus F (a ⊕ b) = F (b∗ a∗ )∗ = (F (b∗ ) F (a∗ ))∗ = F (a= ) ⊕ F (b= ) = F (a) ⊕ F (b). PROPOSITION 5.5.4. If G is an Abelian `-group, then G and D(∆(G)) are isomorphic `-groups. Proof. We denote A = ∆(G) and H = D(A). Then G+ and H+ are isomorphic as latticial monoids, since they are both isomorphic with D(A). If f : G+ → H+ is an isomorphism of latticial monoids, then the function F : G → H defined by F (g) = f (g+ ) − f (g− ) is the unique `-group isomorphism such that F (g) = f (g) for any g ∈ G+ . THEOREM 5.5.5. The functors ∆ and D establish a categorical equivalence between the category Perf of perfect MV-algebras and the category ALG of Abelian `-groups. Proof. By Propositions 5.5.3 and 5.5.4. In the sequel we give a complete characterization for MV-algebras of finite rank. THEOREM 5.5.6. The following are equivalent for any MV-algebra A and n ≥ 1: (a)

A is an MV-algebra of rank n,

(b)

A ' Γ(Z ×lex G, hn, gi) with G Abelian `-group and g ∈ G.

Proof. It is easy to check that Γ(Z ×lex G, hn, gi) has rank n. Let A be an MV-algebra of rank n and let ψ be the isomorphism from A/Rad (A) onto Łn+1 and let {0, b, 2b, . . . , nb} be defined as in Proposition 4.3.24. Further, let G = D(hRad (A)i). We set A0 = Γ(Z ×lex G, hn, gi) with g = [(nb)∗ , 0] and we want to show that A is isomorphic to A0 . Define a map ϕ : A → A0 in the following way: if ψ([x]Rad(A) ) = r/n with 0 ≤ r ≤ n by Lemma 3.2.6 we have x = (rb⊕) µ∗ with , µ ∈ Rad (A). We set ϕ(x) = hr, [, µ]i. Note that in particular, if x ∈ Rad (A), ϕ(x) = h0, [x, 0]i. In order to prove that ϕ is an isomorphism, note that Lemmas 3.2.6 and 3.5.12 insure that ϕ is well-defined and it is a bijection by Proposition 4.3.24. The element 1 ∈ A is such that 1 = nb ⊕ (nb)∗ with (nb)∗ ∈ Rad (A) hence ϕ(1) = hn, [(nb)∗ , 0]i. Let us prove that ϕ((x)∗ ) = ϕ(x)∗ . Let x = (rb ⊕ ) µ∗ , hence x∗ = ((rb⊕) µ∗ )∗ = ((rb µ∗ )⊕)∗ = ((rb)∗ ⊕µ) ∗ . By Proposition 4.3.24, (rb)∗ = (n − r)b ⊕ (nb)∗ hence x∗ = ((n − r)b ⊕ ((nb)∗ ⊕ µ)) ∗ ). So ϕ(x∗ ) =

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hn − r, [(nb)∗ ⊕ µ, ]i. On the other side, ϕ(x)∗ = hr, [, µ]i∗ = hn − r, [(nb)∗ ⊕ µ, ]i hence the claim follows. Let us prove that ϕ(x ⊕ y) = ϕ(x) ⊕ ϕ(y). Let x = (rb ⊕ ) µ∗ and y = (sb ⊕ 0 ) (µ0 )∗ and first consider the case r + s ≤ n. Then by Lemma 3.2.6 and Proposition 4.3.24 x ⊕ y = (rb ⊕ sb ⊕  ⊕ 0 ) (µ ⊕ µ0 ) = ((r + s)b ⊕  ⊕ 0 ) (µ ⊕ µ0 ) hence ϕ(x ⊕ y) = hr + s, [ ⊕ 0 , µ ⊕ µ0 ]i = ϕ(x) ⊕ ϕ(y). If r + s > n, by definition ϕ(x)⊕ϕ(y) = hn, [(nb)∗ , 0]i. Further, since r > n−s, we have x > y ∗ hence x⊕y = 1 and ϕ(x ⊕ y) = hn, [(nb)∗ , 0]i. By Remark 4.3.35, the class Perf is not equational, since it is not closed under direct products. Using Theorem 5.5.5 we will characterize the equational class generated by Perf as the variety generated by Chang’s MV-algebra C. Denote V(Perf) the variety generated by the class of perfect MV-algebras and by V(C) the variety generated by Chang’s MV-algebra. PROPOSITION 5.5.7. V(Perf) = V(C). Proof. Since C is a perfect MV-algebra, it is obvious that V(Perf) contains V(C). In order to prove the other inclusion, it suffices to show that any perfect MV-algebra is in V(C). Let A be a perfect MV-algebra. Then, by Theorem 5.5.5, A is isomorphic with ∆(G) for some Abelian `-group G. Any Abelian `-group is a homomorphic image of a subdirect product of groups isomorphic with Z. Then there exists an Abelian `-group K, an `-group homomorphism h : K → G and a set I such that G = h(K) and K is an `-subgroup of Z I . It is easy to see that ∆(G) = ∆(h)(∆(K)), so A is a homomorphic image of ∆(K). But ∆(K) is an MV-subalgebra of ∆(Z I ), which is an MV-subalgebra of ∆(Z)I . Since ∆(Z) is isomorphic with C (see Remark 4.3.34) it follows that ∆(K) is isomorphic with a subalgebra of CI , so ∆(K) is in V(C). Hence A is in V(C) since it is a homomorphic image of ∆(K). We proved that any perfect MV-algebra is in V(C), so the desired conclusion follows. DEFINITION 5.5.8. An MV-algebra A has an Archimedean radical if for every a, b ∈ Rad (A), na ≤ b for any n ∈ N implies a = 0. PROPOSITION 5.5.9. If G is an Abelian `-group and A = ∆(G), then the following are equivalent: (a)

G is an Archimedean `-group,

(b)

A has an Archimedean radical.

Proof. Since Rad (A) = {h0, gi | g ≥ 0}, the desired result is straightforward. In the following we prove that any MV-algebra A is, up to isomorphism, an interval algebra in the radical of a perfect MV-algebra. LEMMA 5.5.10. Let P be a perfect MV-algebra, G = D(P ) and u ∈ G+ . Then for any latticial monoids isomorphism f : G+ → Rad (P ) the following are equivalent: (a)

f (u) generates Rad (P ),

(b)

u is a strong unit of G.

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Proof. (a) ⇒ (b) If g ∈ G+ then, by hypothesis, there is n ∈ N such that f (g) ≤ nf (u) = f (nu). Since f is an isomorphism we get g ≤ nu. We have proved that for any g ∈ G+ there exists n ∈ N such that g ≤ nu. Hence, u is a strong unit of G. (b) ⇒ (a) Follows similarly. Recall that for an MV-algebra A and for 0 < a in A the interval MV-algebra A(0, a) is defined in Example 2.4.6. PROPOSITION 5.5.11. For every MV-algebra A there is a perfect MV-algebra P and a ∈ P such that Rad (P ) is generated by {a} and A is isomorphic with the interval algebra P (0, a). Proof. If A is an MV-algebra, then A is isomorphic with Γ(G, u) for some Abelian `u-group hG, ui. If we consider P = ∆(G), then, by Theorem 5.5.5, G and D(P ) are isomorphic `-groups and there exists a latticial monoids isomorphism h : G+ → Rad (P ). Since u is a strong unit of G, by Lemma 5.5.10, a = h(u) is a generator for Rad (P ). We will prove that f : Γ(G, u) → P (0, a) defined by f (x) := h(x) for any x ∈ [0, u]G is an MV-algebra isomorphism. If x, y ∈ [0, u]G , then f (x ⊕ y) = f ((x + y) ∧ u) = f (x + y) ∧ f (u) = (f (x) ⊕ f (y)) ∧ a = f (x) ⊕[0,a] f (y), f (u − x) ⊕ f (x) = f (u − x + x) = f (u) = a = a ∨ f (x) = a f (x)∗ ⊕ f (x). Since Rad (P ) is a cancellative monoid, we get f (x∗ ) = f (u − x) = a f (x)∗ = f (x)∗[0,a] . Hence f is an MV-algebra isomorphism and A is isomorphic with the MValgebra P (0, a). 5.6

Representations by ultrapowers

The representation of the MV-algebras as subdirect product of chains is very useful in practice, but it offers few information about their structure. representation theorem asserts that any MV-algebra is isomorphic to an algebra of nonstandard real valued functions. THEOREM 5.6.1. For any MV-algebra A there is an ultrapower ∗ [0, 1] of the MValgebra [0, 1] such that A can be embedded into the product (∗ [0, 1])Spec(A) . Proof. Let Q A be an MV-algebra. By Theorem 4.1.4, A can be embedded into the direct product {A/P | P ∈ Spec(A)}. If P ∈ Spec(A), then A/P is an MV-chain, so A/P can be embedded into a divisible MV-chain DP by Corollary 5.3.1. Now we consider the set F = {DP | P ∈ Spec(A)}. By Theorem 5.3.6 any two MV-algebras from F are elementarily equivalent. Using the joint embedding property [14, Proposition 3.1.4], there is an MV-algebra D such that DP can be elementarily embedded in D for any P ∈ Spec(A). It follows that D is also elementarily equivalent with the MV-algebras of F. But [0, 1] also is elementarily equivalent with the MV-algebras of F, since [0, 1] is a divisible MV-chain. Thus, by Frayne’s Theorem [14, Corollary 4.3.13], there is an ultrapower ∗ [0, 1] of [0, 1] in which D is elementarily embedded. For any P ∈ Spec(A) we get A/P ,→ DP ,→ D ,→ ∗[0, 1]

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and we denote by ιP : A/P ,→ ∗[0, 1] the resulting embedding. Hence, if we define ι : A → (∗ [0, 1])Spec(A) as ι(a) = {ιP (a) | P ∈ Spec(A)}, we get the desired embedding for the MV-algebra A. We will focus on the dependency of non-standard representation of an MV-algebra A on the cardinality of A, showing how to get, for certain classes of MV-algebras, a non-standard representation having as target algebra a unique ultrapower of the MValgebra [0, 1]. Actually, we explore classes of MV-algebras which are representable via algebras of functions from a set X to a fixed regular ultrapower of [0, 1]. From now on, k is an infinite cardinal. Recall that a model A is α-universal if and only if for every model B of power less than α which is elementarily equivalent to A is elementarily embedded in A. THEOREM 5.6.2 ( [14, Theorem 4.3.12]). Let |L| ≤ Q α and D be a ultrafilter which is α-regular. Then, for every model A, the ultrapower D A is α+ -universal. LEMMA 5.6.3. Let A be an MV-algebra and hG, ui an Abelian `u-group such that A ' Γ(G, u). Let k be an infinite cardinal and |A| = k, then |G| = k. Proof. Suppose |A| = k, where k an infinite cardinal. Let a = {ai | i ∈ Z} be a good sequence. Then there is an integer na such that ai = 1, i < −na and ai = 0, i > na . Thus we can identify the good sequence a with a 2na + 1 tuple, < a−na , a−na +1 , . . . , ana −1 , ana >. Call na the index of a. Then for the set Sn of all good sequences of a given index n, we have |Sn | less than or equal to k2n+1 . But |k2n+1 | = k. Thus Sn will have cardinality at most k. The set S of all good sequences can be identified with the union of all Sn . Hence the cardinality of S, will be |S| ≤ k|Z| which is just k since |Z| is denumerable. So the set of all good sequences S we have |S| < k or |S| = k. To see that |S| = k, just consider the good sequences of index 1, S1 . Let a be an element of A and define a0 = a, a1 = a2 = · · · = 0, a−1 = a−2 = · · · = 1. Thus the sequence looks like < . . . , 1, 1, a, 0, 0, . . . >. This is a good sequence as a−2 + a1 = 1 = a−2 a−1 + a0 = 1 + a = 1 = a−1 a0 + a1 = a + 0 = a = a0 a1 + a2 = 0 = a1 , etc. Clearly then |S1 | = |A| = k and so |S| = k. LEMMA 5.6.4. Let G be an Abelian `-group and k be an infinite cardinal such that |G| = k. Then G can be embedded into a divisible Abelian `-group DG such that |DG | = k. Proof. First we show that any Abelian group can be embedded in an Abelian divisible group of cardinality α, where α = max{ℵ0 , |G|}. Indeed, let X be a generating set of G and let F be the free Abelian group with basis X. Obviously, there exists some H ≤ F such that G is isomorphic to F/H. Let D be the direct product of |X| copies of Q. Since F is isomorphic to the direct product of |X| copies of Z we can embed it in D, in natural way. Hence G embeds in the Abelian divisible group D/H. Moreover |D/H| ≤ |X||Q| ≤ α and obviously |G| ≤ |D/H|. On the other hand, since D/H

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is divisible it is infinite and ℵ0 ≤ |D/H|. D/H can be converted into an `-group as follows: call an element h of D/H positive if nh ∈ G+ for some positive integer n; this makes D/H an `-group. THEOREM 5.6.5. For every cardinal k there is an ultrapower Uk of the MV-algebra [0, 1], such that every MV-chain A of cardinality k embeds in Uk via an ultrafilter k-regular over k. Proof. Let A be an infinite MV-chain such that |A| = k and A ' Γ(G, u), then, by Lemma 5.6.3, G is an ordered Abelian group with strong unit u and |G| = k. Hence hG, ui can be embedded into a divisible ordered group DG with strong unit uD , in addition, |DG | = k. Since DG is elementarily equivalent to R, we get that the MVchain Ad = Γ(DG , uD ) is elementarily equivalent to [0, 1], A can be embedded into Ad , i.e. Ad = Γ(DG , uD ) ≡ [0, 1] and A ,→ Ad . Q Now, by Frayne’s Theorem, Ad can be elementarily embedded into an ultrapower F [0, 1] of [0, 1] via an ultrafilter F . Since the cardinality of A is k, then, in the light of the proof ofQFrayne’s Theorem, F can be k-regular over k. Hence A can be embedded into Uk = F [0, 1], which is the desired ultrapower. THEOREM 5.6.6. For every cardinal k there is an ultrapower, Uk , of the MV-algebra [0, 1], obtained via a ultrafilter k-regular over k, such that every MV-algebra A of cardinality k embeds into an MV-algebra of functions from a set to Uk . Proof. Let A be an infiniteQ MV-algebra of cardinality k. Then by Chang’ representation theorem we have A ,→ P ∈Spec(A) A/P . Moreover, |A/P | ≤ k for every prime ideal P of A. Q Let F be a ultrafilter k-regular over k. By Theorem 5.6.2, the ultrapower F [0, 1] is k+ -universal. Then Q for every P ∈ Spec(A) the MV-chain A/P can be embedded into the ultrapower F [0, 1] of the MV-algebra [0, 1]. Q Indeed the ultrafilter F is independent of A/P . Hence A can be embedded into ( F [0, 1])Spec(A) , then we get the representation claimed by the theorem.

6

Łukasiewicz ∞-valued logic

6.1

The syntax of Ł

The language of the propositional calculus Š consists of: denumerable many propositional variables: v1 , . . . , vn , . . . (the set of all the propositional variables will be denoted by V ); logical connectives: → and ¬; and parenthesis: ( and ). The formulas are defined inductively as follows: (f1)

every propositional variable is a formula,

(f2)

if ϕ is a formula, then ¬ϕ is a formula,

(f3)

if ϕ and ψ are formulas, then (ϕ → ψ) is a formula,

(f4)

a string of symbols is a formula of Š iff it can be shown to be a formula by a finite number of applications of (f1), (f2), and (f3).

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We will denote by Fm Š the set of all formulas of Š. The particular four axiom schemes of this propositional calculus are: (A1)

ϕ → (ψ → ϕ),

(A2)

(ϕ → ψ) → ((ψ → χ) → (ϕ → χ)),

(A3)

((ϕ → ψ) → ψ) → ((ψ → ϕ) → ϕ),

(A4)

(¬ψ → ¬ϕ) → (ϕ → ψ).

The deduction rule is modus ponens (MP): from ϕ and ϕ → ψ infer ψ. DEFINITION 6.1.1 (Syntactic consequences, theorems). Let Θ be a set of formulas and ϕ a formula. A Θ-proof for ϕ is a finite sequence of formulas ϕ1 , . . . , ϕn = ϕ such that, for any i ∈ {1, . . . , n}, at least one of the following conditions holds: (c1) ϕi is an axiom, (c2) ϕi ∈ Θ, (c3)

there are j, k < i such that ϕk is ϕj → ϕi (the formula ϕi follows from ϕj and ϕk using modus ponens).

We will say that ϕ is a syntactic consequence of Θ (or ϕ is provable from Θ) if there exists a Θ-proof for ϕ. We write Θ ` ϕ. The set of all the syntactic consequences of Θ will be denoted by Theor (Θ). A formula ϕ will be called a theorem (or provable formula) if it is provable from the empty set. This will be denoted by ` ϕ. In this case, a proof for ϕ will be a sequence of formulas ϕ1 , . . . , ϕn = ϕ such that for any i ∈ {1, . . . , n}, one of the above conditions (c1) or (c3) is satisfied. The set of all the theorems will be denoted by Theor . LEMMA 6.1.2 (Syntactic compactness). If Θ is a set of formulas and ϕ is a formula such that Θ ` ϕ, then Γ ` ϕ for some finite subset Γ ⊆ Θ. Proof. If Θ ` ϕ, then there exists a Θ-proof ϕ1 , . . . , ϕn = ϕ for ϕ. We consider Γ = Θ ∩ {ϕ1 , . . . , ϕn }. It is straightforward by Definition 6.1.1 that ϕ1 , . . . , ϕn = ϕ is a Γ-proof for ϕ. In the sequel we will prove some theorems of Š. We present a detailed proof for the theorem (1)

ϕ → ((ϕ → ψ) → ψ).

The proof is: ` ϕ → ((ψ → ϕ) → ϕ) ` (ϕ → ((ψ → ϕ) → ϕ)) → (((ψ → ϕ) → ϕ) → ((ϕ → ψ) → ψ)) → (ϕ → ((ϕ → ψ) → ψ))) ` (((ψ → ϕ) → ϕ) → ((ϕ → ψ) → ψ)) → (ϕ → ((ϕ → ψ) → ψ)) ` ((ψ → ϕ) → ϕ) → ((ϕ → ψ) → ψ) ` ϕ → ((ϕ → ψ) → ψ)

(A1) (A2) (MP) (A3) (MP)

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In the sequel, when we will use an axiom (or a previous proved theorem), like (A2) in the previous proof, we will not write the axiom (or the theorem) obtained by making some particular substitution. We will only indicate the axiom (theorem) used and how many time we apply modus ponens. For example, the previous proof will be: ` ϕ → ((ψ → ϕ) → ϕ) ` ((ψ → ϕ) → ϕ) → ((ϕ → ψ) → ψ) ` ϕ → ((ϕ → ψ) → ψ)

(A1) (A3) (A2), 2 (MP)

PROPOSITION 6.1.3. In Š the following formulas are theorems: (2)

(ϕ → (ψ → χ)) → (ψ → (ϕ → χ)),

(3)

ϕ → ϕ,

(4)

(ψ → χ) → ((ϕ → ψ) → (ϕ → χ)),

(5)

¬¬ϕ → (ψ → ϕ),

(6)

¬¬ϕ → ϕ,

(7)

(ϕ → ¬ψ) → (ψ → ¬ϕ),

(8)

ϕ → ¬¬ϕ,

(9)

(¬ϕ → ψ) → (¬ψ → ϕ),

(10) (ϕ → ψ) → (¬ψ → ¬ϕ). Proof. Let α be an axiom and θ the formula ((ψ → χ) → χ) → (ϕ → χ) then (2)

` (ψ → ((ψ → χ) → χ)) → (θ → (ψ → (ϕ → χ))) ` θ → (ψ → (ϕ → χ)) ` (ϕ → (ψ → χ)) → θ ` (ϕ → (ψ → χ)) → (ψ → (ϕ → χ))

(A2) (1), (MP) (A2) (A2), 2 (MP)

(3)

` ϕ → (α → ϕ) ` (ϕ → (α → ϕ)) → (α → (ϕ → ϕ)) ` α → (ϕ → ϕ) `α `ϕ→ϕ

(A1) (2) (MP) axiom (MP)

(4)

` (ψ → χ) → ((ϕ → ψ) → (ϕ → χ))

(A1), (2), (MP)

(5)

` ¬¬ϕ → (¬¬ψ → ¬¬ϕ) ` (¬¬ψ → ¬¬ϕ) → (¬ϕ → ¬ψ) ` ¬¬ϕ → (¬ϕ → ¬ψ) ` (¬ϕ → ¬ψ) → (ψ → ϕ) ` ¬¬ϕ → (ψ → ϕ)

(A1) (A4) (A2), 2 (MP) (A4) (A2), 2 (MP)

(6)

` ¬¬ϕ → (α → ϕ) ` α → (¬¬ϕ → ϕ) `α ` ¬¬ϕ → ϕ

(5) (2), (MP) axiom (MP)

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(7)

` ¬¬ϕ → ϕ ` (ϕ → ¬ψ) → (¬¬ϕ → ¬ψ) ` (¬¬ϕ → ¬ψ) → (ψ → ¬ϕ) ` (ϕ → ¬ψ) → (ψ → ¬ϕ)

(6) (A2), (MP) (A4) (A2), 2 (MP)

(8)

` ¬ϕ → ¬ϕ ` (¬ϕ → ¬ϕ) → (ϕ → ¬¬ϕ) ` ϕ → ¬¬ϕ

(3) (7) (MP)

(9)

` ψ → ¬¬ψ ` (¬ϕ → ψ) → (¬ϕ → ¬¬ψ) ` (¬ϕ → ¬¬ψ) → (¬ψ → ϕ) ` (¬ϕ → ψ) → (¬ψ → ϕ)

(8) (4), (MP) (A4) (A2), 2 (MP)

` ¬¬ϕ → ϕ (6) ` (ϕ → ψ) → (¬¬ϕ → ψ) (A2), (MP) ` (¬¬ϕ → ψ) → (¬ψ → ¬ϕ) (9) ` (ϕ → ψ) → (¬ψ → ¬ϕ) (A2), 2 (MP) We define other logical connectives as follows (for any n ≥ 1): (10)

ϕ ⊕ ψ := ¬ϕ → ψ ϕ ∨ ψ := (ϕ → ψ) → ψ ϕn := ϕ · · · ϕ {z } | n



ϕ ψ := ¬(ϕ → ¬ψ), ϕ ∧ ψ := ϕ (ϕ → ψ), nϕ := ϕ ⊕ · · · ⊕ ϕ | {z } n

One can easily see that (A3) is equivalent to ϕ ∨ ψ → ψ ∨ ϕ and (1) is equivalent to ϕ → ϕ ∨ ψ. PROPOSITION 6.1.4. The following formulas are theorems of Š: (11) (ϕ → (ψ → χ)) → ((ϕ ψ) → χ), (12) ((ϕ ψ) → χ) → (ϕ → (ψ → χ)), (13) (ϕ ψ) → (ψ ϕ), (14) ϕ → (ψ → (ψ ϕ)), (15) ϕ → (ψ → (ϕ ψ)), (16) (ϕ ¬ϕ) → ψ, (17) (ϕ ψ) → ϕ, (18) (ϕ → ψ) → ((χ ϕ) → (χ ψ)), (19) ((χ ϕ) → (χ ψ)) → ((ϕ χ) → (ψ χ)), (20) (ϕ (ϕ → ψ)) → ψ, (21) (ϕ → ψ) → ((ϕ ∨ χ) → (ψ ∨ χ)), (22) ((ϕ ∨ χ) → (ψ ∨ χ)) → ((χ ∨ ϕ) → (χ ∨ ψ)), (23) (ϕ ∨ ϕ) → ϕ. Proof. The following sequences are formal proofs in Š.

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(11)

` (ψ → χ) → (¬χ → ¬ψ) ` (ϕ → (ψ → χ)) → (ϕ → (¬χ → ¬ψ)) ` (ϕ → (¬χ → ¬ψ)) → (¬χ → (ϕ → ¬ψ)) ` (ϕ → (ψ → χ)) → (¬χ → (ϕ → ¬ψ)) ` (¬χ → (ϕ → ¬ψ)) → (¬(ϕ → ¬ψ) → χ) ` (ϕ → (ψ → χ)) → ((ϕ ψ) → χ)

(10) (4), (MP) (2) (A2), 2 (MP) (9) (A2), 2 (MP)

(12)

` ((ϕ ψ) → χ) → (¬χ → (ϕ → ¬ψ)) ` (¬χ → (ϕ → ¬ψ)) → (ϕ → (¬χ → ¬ψ)) ` ((ϕ ψ) → χ) → (ϕ → (¬χ → ¬ψ)) ` (ϕ → (¬χ → ¬ψ)) → (ϕ → (ψ → χ)) ` ((ϕ ψ) → χ) → (ϕ → (ψ → χ)) ` (ψ → ¬ϕ) → (ϕ → ¬ψ) ` ¬(ϕ → ¬ψ) → ¬(ψ → ¬ϕ) ` (ϕ ψ) → (ψ ϕ)

(9) (2) (A2), 2 (MP) (A4), (4) (A2), 2 (MP) (7) (10)

(13)

(14)

follows from (13) and (12) using (MP)

(15)

follows from (14) and (2) using (MP)

(16)

` ϕ → (¬ψ → ϕ) ` (¬ψ → ϕ) → (¬ϕ → ψ) ` ϕ → (¬ϕ → ψ) ` (ϕ ¬ϕ) → ψ

(A1) (9) (A2), 2 (MP) (11), (MP)

(17)

` (ϕ ¬ϕ) → ¬ψ ` ϕ → (¬ϕ → ¬ψ) ` ¬ϕ → (ϕ → ¬ψ) ` ¬(ϕ → ¬ψ) → ϕ ` (ϕ ψ) → ϕ

(16) (12), (MP) (2), (MP) (9), (MP)

(18)

` (ϕ → ψ) → (¬ψ → ¬ϕ) ` (¬ψ → ¬ϕ) → ((χ → ¬ψ) → (χ → ¬ϕ)) ` (ϕ → ψ) → ((χ → ¬ψ) → (χ → ¬ϕ)) ` ((χ → ¬ψ) → (χ → ¬ϕ)) → ((χ ϕ) → (χ ψ)) ` (ϕ → ψ) → ((χ ϕ) → (χ ψ))

(10) (4) (A2), 2 (MP) (10) (A2), 2 (MP)

(19)

` (ϕ χ) → (χ ϕ) ` ((χ ϕ) → (χ ψ)) → ((ϕ χ) → (χ ψ)) ` (χ ψ) → (ψ χ) ` ((ϕ χ) → (χ ψ)) → ((ϕ χ) → (ψ χ)) ` ((χ ϕ) → (χ ψ)) → ((ϕ χ) → (ψ χ))

(13) (A2), (MP) (13) (A4), (MP) (A2), 2 (MP)

(20)

` (ϕ → ψ) → (ϕ → ψ) ` (ϕ (ϕ → ψ)) → ψ

(3) (11), (MP)

(21)

` (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)) ` ((ψ → χ) → (ϕ → χ)) → (((ϕ → χ) → χ) → ((ψ → χ) → χ)) ` (ϕ → ψ) → (((ϕ → χ) → χ) → ((ψ → χ) → χ)) ` (ϕ → ψ) → ((ϕ ∨ χ) → (ψ ∨ χ))

(A2) (A2) (A2), 2 (MP)

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(22)

is similar with (19), but we use (A3) instead of (13).

(23)

` (ϕ → ϕ) → ((ϕ → (ϕ → ϕ)) → (ϕ → ϕ)) ` (ϕ → (ϕ → ϕ)) → (ϕ → ϕ) ` ((ϕ → ϕ) → ϕ) → ϕ ` (ϕ ∨ ϕ) → ϕ

(A1) (3), (MP) (A3), (MP) 

LEMMA 6.1.5. Let Θ be a set of formulas such that Θ ` ϕ → ψ and Θ ` α → β. Then: (a)

Θ ` (ϕ α) → (ψ β),

(b)

Θ ` (ϕ ∨ α) → (ψ ∨ β).

Proof. (a) The following sequence is formal proof in Š. Θ`ϕ→ψ Θ ` (α ϕ) → (α ψ) (18), (MP) Θ ` (ϕ α) → (ψ α) (19), (MP) Θ`α→β Θ ` (ψ α) → (ψ β) (18), (MP) Θ ` (ϕ α) → (ψ β) (A2), 2 (MP) (b) Similar to (a), but we use (21) instead of (18) and (22) instead of (19). LEMMA 6.1.6. If ϕ is a formula and Θ is a set of formulas, then: (a)

` ϕn → ϕ for any n ≥ 1,

(b)

Θ ` ϕ implies Θ ` ϕn for any n ≥ 1,

(c)

if Θ ` ϕn for some n ≥ 1, then Θ ` ϕ.

Proof. (a) If n = 1, then ` ϕ → ϕ by (3). If n > 1, we get ` ϕ ϕn−1 → ϕ by (17). (b) We prove that Θ ` ϕn for any n ≥ 1 by induction. If n = 1, then the intended result follows by (3). Now we suppose that Θ ` ϕn . It follows that Θ ` ϕn+1 → ϕn+1 (3) Θ ` (ϕn ϕ) → ϕn+1 Θ ` ϕn → (ϕ → ϕn+1 ) (12), (MP) Θ ` ϕn induction hypothesis Θ`ϕ Θ ` ϕn+1 2 (MP) (c) Follows by (a). THEOREM 6.1.7 (Implicational deduction theorem). If Θ ⊆ Fm Š and ϕ, ψ ∈ Fm Š , then Θ ∪ {ϕ} ` ψ iff there is n ≥ 1 such that Θ ` ϕn → ψ. Proof. If Θ ` ϕn → ψ for some n ≥ 1, then Θ ∪ {ϕ} ` ϕ Θ ∪ {ϕ} ` ϕn Θ ∪ {ϕ} ` ϕn → ψ Θ ∪ {ϕ} ` ψ

Lemma 6.1.6 (b) (Θ ⊆ Θ ∪ {ϕ}) (MP)

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Conversely, suppose that Θ∪{ϕ} ` ψ and let α1 , . . . , αk = ψ be a Θ∪{ϕ}-proof for ψ. We prove by induction that, for any 1 ≤ i ≤ k, there is ni ≥ 1 such that Θ ` ϕni → αi . For i = 1 we have three possible cases: – if α1 is an axiom, then Θ ` α1 → (ϕ → α1 ) (A1) Θ ` α1 axiom Θ ` ϕ → α1 (MP) – if α1 ∈ Θ the proof is similar, – if α1 = ϕ, then Θ ` ϕ → ϕ by (3). Thus, if i = 1, then n1 = 1. Let 1 < i ≤ k and suppose the desired result holds for j < i. If αi is an axiom or αi ∈ Θ ∪ {ϕ}, then we get ni = 1 like above. Suppose that αi is follows from αj and αt using (MP), where j, t < i. By induction, there are nj , nt ≥ 1 such that Θ ` ϕnj → αj and Θ ` ϕnt → αt . We assume that αt = αj → αi . It follows that Θ ` ϕnj → αj and Θ ` ϕnt → (αj → αi ). By Lemma 6.1.5 (a), we get Θ ` (ϕnj ϕnt ) → (αj (αj → αi )). We consider ni = nj + nt . Thus, Θ ` ϕni → (αj (αj → αi )) Θ ` (αj (αj → αi )) → αi Θ ` ϕni → αi 6.2

(20) (A2), 2 (MP)



The Lindenbaum–Tarski algebra (Ł(Θ), Ł) In the sequel Θ is a fixed set of formulas. For any two formulas ϕ and ψ we define ϕ ∼Θ ψ

iff Θ ` ϕ → ψ and Θ ` ψ → ϕ.

LEMMA 6.2.1. The relation ∼Θ is an equivalence relation on Fm Š . Proof. The relation ∼Θ is reflexive by (3) and it is symmetric by definition. In order to prove the transitivity, we suppose that ϕ ∼Θ ψ and ψ ∼Θ χ. It follows that Θ ` ϕ → ψ and Θ ` ψ → χ. Using (A2) and applying twice (MP), we infer that Θ ` ϕ → χ. Similarly, it follows that Θ ` χ → ϕ, so ϕ ∼Θ χ. For any formula ϕ ∈ Fm Š we will denote by [ϕ]Θ the equivalence class of ϕ with respect to ∼Θ , i.e. [ϕ]Θ = {ψ | Θ ` ϕ → ψ and Θ ` ψ → ϕ}. Consequently, Fm Š /∼Θ = {[ϕ]Θ | ϕ ∈ Fm Š } is the quotient of Fm Š with respect to ∼Θ . LEMMA 6.2.2. For any formulas ϕ and ψ the following properties hold: (a)

Θ ` ϕ iff [ϕ]Θ = Theor (Θ),

(b)

if Θ ` ϕ, then [ϕ → ψ]Θ = [ψ]Θ .

Proof. (a) We suppose that Θ ` ϕ and we consider ψ ∈ Theor (Θ). It follows that Θ ` ψ. By (A1), we get Θ ` ψ → (ϕ → ψ) and Θ ` ϕ → (ψ → ϕ). Using (MP), we infer that Θ ` ψ → ϕ and Θ ` ϕ → ψ, so Theor (Θ) ⊆ [ϕ]Θ . Conversely, if ψ ∈ [ϕ]Θ , then Θ ` ϕ → ψ. Using again (MP), we get Θ ` ψ, so ψ ∈ Theor (Θ). We proved that [ϕ]Θ ⊆ Theor (Θ). Now we suppose that [ϕ]Θ = Theor (Θ). Let α be an axiom. Then α ∈ Theor (Θ) and α ∼Θ ϕ. We get Θ ` α and Θ ` α → ϕ, so Θ ` ϕ by (MP).

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(b) By (A1), it follows that Θ ` ψ → (ϕ → ψ) and Θ ` ϕ → ((ψ → ϕ) → ϕ). By hypothesis and (MP), we infer that Θ ` (ψ → ϕ) → ϕ. Using (A3) and (MP), it follows that Θ ` (ϕ → ψ) → ψ. Thus Θ ` ψ → (ϕ → ψ) and Θ ` (ϕ → ψ) → ψ, so [ϕ → ψ]Θ = [ψ]Θ . On Fm Š /∼Θ we define the following operations: [ϕ]∗Θ := [¬ϕ]Θ

[ϕ]Θ → [ψ]Θ := [ϕ → ψ]Θ

[ϕ]Θ ⊕ [ψ]Θ := [(¬ϕ) → ψ]Θ .

PROPOSITION 6.2.3. The structure hFm Š /∼Θ , →, ∗ , Theor (Θ)i is a Wajsberg algebra. Proof. If α is an axiom, then Θ ` α so, by Lemma 6.2.2 (a), [α]Θ = Theor (Θ). Thus the Wajsberg algebras axioms (W2)–(W4) from Definition 2.3.6 follows by Lemma 6.2.2 (a) applied to axioms (A2)–(A4). The axiom (W1) follows by Lemma 6.2.2 (b). COROLLARY 6.2.4. The structure hFm Š /∼Θ , ⊕, ∗ , Theor (Θ)∗ i is an MV-algebra. Proof. It is straightforward by Proposition 2.3.10. In the sequel, we will denote Ł(Θ) = hFm Š /∼Θ , ⊕, ∗ , Theor (Θ)∗ i and this MValgebra will be called the Lindenbaum–Tarski algebra of Θ. Note that, for any set of formulas Θ, the Lindenbaum–Tarski algebras of Θ and Theor (Θ) coincide, i.e. Ł(Θ) = Ł(Theor (Θ)). In the particular case when Θ = ∅, the relation ∼∅ will be simply denoted by ∼. Thus, we have ϕ∼ψ

iff

` ϕ → ψ and ` ψ → ϕ.

The equivalence class of a formula ϕ with respect to ∼ will be denoted by [ϕ]. Consequently, the MV-algebra Ł(∅) = Ł(T heor) will be denoted by Ł and this is the Lindenbaum–Tarski algebra of the ∞-valued Łukasiewicz propositional calculus. In order to simplify the notation, Ł(Θ) will denote both the Lindenbaum–Tarski algebra and its support. 6.3

Consistent sets, linear sets, and deductive systems

DEFINITION 6.3.1. Let Θ be a set of formulas. We say that Θ is consistent if there is a formula ϕ such that Θ 6` ϕ. Otherwise, Θ is called inconsistent. One can easily see that Θ is inconsistent iff Theor (Θ) = Fm Š . The set Θ is maximally consistent if it cannot be strictly included in any consistent set. LEMMA 6.3.2. For a set Θ of formulas the following are equivalent: (a)

Θ is inconsistent,

(b)

Θ ` ϕ and Θ ` ¬ϕ for some ϕ ∈ Fm Š ,

(c) Θ ` ϕ ¬ϕ for some ϕ ∈ Fm Š . Proof. (a) ⇒ (b) Straightforward, since Theor (Θ) = Fm Š . (b) ⇒ (c) By hypothesis, (15) and twice (MP). (c) ⇒ (a) By hypothesis and (16), using (MP), we get Θ ` ψ for any formula ψ.

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COROLLARY 6.3.3. Any consistent set of formulas is contained in a maximally consistent set of formulas. Proof. Let Θ be a consistent set of formulas and T = {Γ ⊆ Fm Š | Γ is consistent and Θ ⊆ Γ}. We will prove that T is inductively ordered. Thus, the desired conclusion follows by Zorn’s Lemma. Let {Γi | i ∈ I} be a totally ordered (with respect to the set inclusion) family from T . WeS have to prove that this family has an upper bound in T . This upper bound will be Γ = {Γi | i ∈ I}. We only have to prove that Γ is consistent. Suppose that Γ is inconsistent. By Lemma 6.3.2, we get Γ ` ϕ ¬ϕ. Than there exists a Γ-proof ϕ1 , . . . , ϕn for ϕ ¬ϕ. Let i1 , . . . , in ∈ I such that ϕj ∈ Γij for any j ∈ {1, . . . , n}. Since the family {Γi | i ∈ I} is totally ordered, there is k ∈ {1, . . . , n} such that Γij ⊆ Γik , for any j ∈ {1, . . . , n}. Thus ϕ1 , . . . , ϕn is a Γik -proof for ϕ ¬ϕ so, by Lemma 6.3.2, Γik is inconsistent, which is a contradiction. Hence, Γ is consistent and T is inductively ordered. LEMMA 6.3.4. If Θ is a set of formulas and ϕ is a formula, then the following are equivalent: (a)

Θ ∪ {ϕ} is inconsistent,

(b)

Θ ` ¬(ϕn ) for some n ≥ 1.

Proof. (a) ⇒ (b) Let α be any axiom. By hypothesis, Θ ∪ {ϕ} ` ¬α so, by Theorem 6.1.7, there is n ≥ 1 such that Θ ` ϕn → ¬α. Using (7) and (MP), we get Θ ` α → ¬(ϕn ). Since α is a theorem, Θ ` ¬(ϕn ). (b) ⇒ (a) Let n ≥ 1 such that Θ ` ¬(ϕn ), then Θ∪{ϕ} ` ¬(ϕn ). By Lemma 6.1.6 (b), it follows that Θ ∪ {ϕ} ` ϕn for any n ≥ 1. Hence, using (15) and twice (MP), we get Θ ∪ {ϕ} ` ϕn ¬(ϕn ). The desired conclusion follows by Lemma 6.3.2. LEMMA 6.3.5. If Θ is a maximally consistent set of formulas, then Θ = Theor (Θ). Proof. Let ϕ be a formula such that ϕ ∈ Theor (Θ) and ϕ ∈ / Θ. Since Θ is maximally consistent, we infer that Θ ∪ {ϕ} is inconsistent. By Lemma 6.3.4, there is n ≥ 1 such that Θ ` ¬(ϕn ). Because Θ ` ϕ, by Lemma 6.1.6 (b), it follows that Θ ` ϕn . By Lemma 6.3.2, Θ is inconsistent, which is a contradiction. Thus, Theor (Θ) ⊆ Θ. Since Θ ⊆ Theor (Θ) for any set of formulas, we get Θ = Theor (Θ). LEMMA 6.3.6. For a consistent set of formulas Θ, the following are equivalent: (a)

Θ is maximally consistent,

(b)

if ϕ ∈ Fm Š and ϕ ∈ / Θ, then ¬(ϕn ) ∈ Θ for some n ≥ 1.

Proof. (a) ⇒ (b) If ϕ ∈ / Θ, then Θ ∪ {ϕ} is inconsistent. By Lemma 6.3.4, there is n ≥ 1 such that Θ ` ¬(ϕn ). Using Lemma 6.3.5, this means that ¬(ϕn ) ∈ Θ. (b) ⇒ (a) By hypothesis and Lemma 6.3.4, we infer that Θ ∪ {ϕ} is inconsistent for any ϕ ∈ / Θ. It is obvious that Θ is maximally consistent.

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DEFINITION 6.3.7. A set of formulas Θ is called linear if Θ ` ϕ → ψ or Θ ` ψ → ϕ for any two formulas ϕ and ψ. LEMMA 6.3.8. For any n ≥ 1, the following formula is a theorem of L: (24) (ϕ → ψ)n ∨ (ψ → ϕ)n . Proof. By Lemma 2.3.1 (e), the identity (a → b)n ∨ (b → a)n = 1 holds in any MV-algebra, so it holds in the Lindenbaum–Tarski algebra Ł. Hence, for any two formulas ϕ and ψ, we get ([ϕ] → [ψ])n ∨ ([ψ] → [ϕ])n = T heor

[(ϕ → ψ)n ∨ (ψ → ϕ)n ] = T heor.

and

The desired conclusion follows by Lemma 6.2.2 (a). PROPOSITION 6.3.9. For any consistent set of formulas Θ and formula ϕ such that Θ 6` ϕ, there is a consistent and linear set of formulas Γ such that Θ ⊆ Γ and Γ 6` ϕ. Proof. We inductively define a sequence {Γn | n ∈ N} with the following properties: - Θ ⊆ Γn for any n ∈ N, - Γn 6` ϕ for any n ∈ N, - for any two formulas ψ and χ there is n ∈ N such that Γn ` ψ→χ or Γn ` χ→ψ, - Γn ⊆ Γn+1 for any n ∈ N. Let {(ψn , χn ) | n ∈ N} be an enumeration for Fm Š × Fm Š . We define the sequence {Γn | n ∈ N} as follows. For n = 0 we set Γ0 = Θ. Now we suppose that Γn is defined having the above properties. We prove that (∗)

Γn ∪ {ψn → χn } 6` ϕ or

Γn ∪ {χn → ψn } 6` ϕ.

If Γn ∪ {ψn → χn } ` ϕ and Γn ∪ {χn → ψn } ` ϕ then, by Theorem 6.1.7, there exist two natural numbers r and s such that Γn ` (ψn → χn )r → ϕ and Γn ` (χn → ψn )s → ϕ. If k = max{r, s}, then, using (17) and (A2), one can easily prove that Γn ` (ψn → χn )k → ϕ and

Γn ` (χn → ψn )k → ϕ.

Using Lemma 6.1.5 (b), we infer that Γn ` ((ψn → χn )k ∨ (χn → ψn )k ) → ϕ ∨ ϕ. By (23) and (A2), using twice (MP), we get Γn ` ((ψn → χn )k ∨ (χn → ψn )k ) → ϕ. From Lemma 6.3.8 it follows that Γn ` ϕ, which is a contradiction. Hence, the condition (*) is proved. We define ( Γn ∪ {ψn → χn } if Γn ∪ {ψn → χn } 6` ϕ, Γn+1 = Γn ∪ {χn → ψn } otherwise.

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S It is obvious that Γn+1 satisfies the required conditions. Let Γ = {Γn | n ∈ N}. It is obvious that Θ ⊆ Γ. We also have that Γ 6` ϕ, since otherwise, by Lemma 6.1.2, there would be a finite subset ∆ ⊆ Γ such that ∆ ` ϕ. Since ∆ is finite, then there is some n ∈ N such that ∆ ⊆ Γn , so Γn ` ϕ which is a contradiction. Thus, Γ is also consistent. We only have to prove that Γ is linear. If ψ and χ are two arbitrary formulas, then there is some n ∈ N such that (ψ, χ) = (ψn , χn ) and ψ → χ ∈ Γn+1 ⊆ Γ or χ → ψ ∈ Γn+1 ⊆ Γ. COROLLARY 6.3.10. Any maximally consistent set is linear. Proof. Let Θ be a maximally consistent set. By Proposition 6.3.9, there is a consistent and linear set of formulas, Γ, such that Θ ⊆ Γ. Since Θ is maximally consistent, we get Θ = Γ. Hence, Θ is linear. DEFINITION 6.3.11. We call deductive system a set Θ of formulas which contains the axioms and it is closed with respect to (MP). PROPOSITION 6.3.12. If Θ is a set of formulas, then the following are equivalent: (a)

Θ is a deductive system,

(b)

Θ = Theor (Θ).

Proof. (a) ⇒ (b) We have to prove that Theor (Θ) ⊆ Θ. If ϕ ∈ Theor (Θ), then Θ ` ϕ. Let ϕ1 , . . . , ϕn = ϕ be a Θ-proof for ϕ. We prove by induction that ϕi ∈ Θ for any i ∈ {1, . . . , n}. For i = 0, we have ϕ0 ∈ Θ or ϕ0 is an axiom. In this case we also get ϕ0 ∈ Θ, since Θ is a deductive system. We suppose that ϕj ∈ Θ for any j < i and we prove that ϕi ∈ Θ. If ϕi is an axiom or ϕi ∈ Θ, then the desired conclusion is straightforward. Otherwise, there are k, j < i such that ϕk = ϕj → ϕi . By induction, ϕk and ϕj are in Θ. Since Θ is closed with respect to (MP), we get ϕi ∈ Θ. Hence, ϕ is in Θ, so Θ = Theor (Θ). (b) ⇒ (a) If α is an axiom, then Θ ` α, so α ∈ Theor (Θ) = Θ. If ϕ and ϕ → ψ are in Θ = Theor (Θ), then Θ ` ϕ and Θ ` ϕ → ψ. By (MP) we get Θ ` ψ, so ψ ∈ Theor (Θ) = Θ. COROLLARY 6.3.13. Any maximally consistent set of formulas is a deductive system. Proof. By Lemma 6.3.5 and Proposition 6.3.12. In the sequel, we will provide an algebraic interpretation for the particular sets of formulas presented above. If Θ ⊆ Fm Š and F ⊆ Ł then we define [Θ] := {[ϕ] | ϕ ∈ Θ}

and

F ∗ := {ϕ | [ϕ] ∈ F }.

The following result will be very useful. LEMMA 6.3.14. If Θ is a deductive system and ϕ is a formula, then the following are equivalent: (a)

ϕ ∈ Θ,

(b)

[ϕ] ∈ [Θ].

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Proof. (a) ⇒ (b) Straightforward by the definition of [Θ]. (b) ⇒ (a) If [ϕ] ∈ [Θ], then there is ψ ∈ Θ such that [ϕ] = [ψ]. It follows that ψ ∈ Θ and ` ψ → ϕ. Since Θ is a deductive system, we get ϕ ∈ Θ. We recall that, in an MV-algebra, the notions of filter, prime filter and ultrafilter (maximal filter) are the dual notions of those of ideal, prime ideal and maximal ideal, respectively. PROPOSITION 6.3.15. If Θ is a deductive system and F is a filter of Ł, then the following properties hold: (a)

[Θ] is a filter in Ł,

(b)

F ∗ is a deductive system,

(c) [Θ]∗ = Θ, (d)

[F ∗ ] = F ,

(e) Θ is maximally consistent iff [Θ] is an ultrafilter. Proof. One can see Definition 2.7.4 and Proposition 2.7.3 (a) for the notion of filter. (a) If α is an axiom, then α ∈ Θ, so [α] = T heor ∈ [Θ]. Thus, [Θ] is a nonempty set of Ł. If [ϕ] and [ϕ] → [ψ] = [ϕ → ψ] are in [Θ], then ϕ and ϕ → ψ are in Θ by Lemma 6.3.14. Since Θ is a deductive system, we get ψ ∈ Θ. Thus, [ψ] ∈ [Θ] and [Θ] is a filter in Ł. (b) Since F is a filter and Theor is the last element of Ł, it follows that Theor ∈ F . If α is an axiom, then [α] = T heor by Lemma 6.2.2 (a), so α ∈ F ∗ . Hence, F ∗ contains the axioms. We have to prove that it is closed with respect to modus ponens. If ϕ and ϕ → ψ are in F ∗ , then [ϕ] and [ϕ → ψ] = [ϕ] → [ψ] are in F . By Proposition 2.7.3 (a), we get [ψ] ∈ F , so ψ ∈ F ∗ . We proved that F ∗ is a deductive system. (c) If ϕ is a formula then, using Lemma 6.3.14, it follows that: ϕ ∈ [Θ]∗ (d)

iff [ϕ] ∈ [Θ] iff ϕ ∈ Θ.

Let ϕ be a formula. Using (b) and Lemma 6.3.14 we get: [ϕ] ∈ [F ∗ ] iff ϕ ∈ F ∗

iff [ϕ] ∈ F.

(e) Firstly, we have ϕ ∈ / Θ iff [ϕ] ∈ / [Θ] by Lemma 6.3.14. Thus, Θ is consistent iff [Θ] is a proper filter. Let Θ be a maximal consistent set of formulas and suppose that [Θ] ⊆ F , where F is a proper filter in Ł. By (c), we infer Θ = [Θ]∗ ⊆ F ∗ and that F ∗ is consistent. Thus, Θ = F ∗ and [Θ] = [F ∗ ] = F by (d). We proved that [Θ] is an ultrafilter. The converse implication follows similarly. COROLLARY 6.3.16. There is an order isomorphism between the set of all the deductive systems of Š and the set of all the filters of Ł (ordered by set theoretical inclusion). Moreover, the maximally consistent sets correspond to ultrafilters. Proof. By Proposition 6.3.15.

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PROPOSITION 6.3.17. Let Θ be a set of formulas. It follows that: (a)

Θ is consistent iff [Theor (Θ)] is a proper filter,

(b)

Θ is consistent and linear iff [Theor (Θ)] is a prime filter.

Proof. By Proposition 6.3.12, Theor (Θ) is a deductive system, therefore [Theor (Θ)] is a filter of Ł. (a) If Θ is consistent, then there is a formula ϕ such that Θ 6` ϕ. It follows that ϕ∈ / Theor (Θ). By Lemma 6.3.14, we get [ϕ] ∈ / [Theor (Θ)], so [Theor (Θ)] is proper. The converse implication is straightforward. (b) Let Θ be a consistent and linear set of formulas. By (a), [Theor (Θ)] is a proper filter. Since Θ is linear, we get Θ ` ϕ → ψ or Θ ` ψ → ϕ for any two formulas ϕ and ψ. It follows that ϕ → ψ ∈ Theor (Θ) or ψ → ϕ ∈ Theor (Θ). Hence, in Ł we get [ϕ] → [ψ] ∈ [Theor (Θ)] or [ψ] → [ϕ] ∈ [Theor (Θ)], which is the dual of condition (a) from Proposition 3.3.1. It follows that [Theor (Θ)] is a prime filter. The converse implication can be proved similarly. LEMMA 6.3.18. If Θ is a set of formulas and F = [Theor (Θ)], then the following are equivalent for any two formulas ϕ and ψ: (a)

ϕ ∼Θ ψ,

(b)

[ϕ] ∼F [ψ],

where ∼F is the congruence relation associated to the filter F in Ł. Proof. (a) ⇒ (b) If ϕ ∼Θ ψ, then Θ ` ϕ → ψ and Θ ` ψ → ϕ. It follows that [ϕ] → [ψ] ∈ F and [ψ] → [ϕ] ∈ F . By Remark 3.2.13, this means that [ϕ] ∼F [ψ]. (b) ⇒ (a) Straightforward using Lemma 6.3.14. COROLLARY 6.3.19. Let Θ be a set of formulas. Then Ł(Θ) and the quotient algebra Ł/[Theor (Θ)] are isomorphic MV-algebras. Proof. It follows directly from Lemma 6.3.18. COROLLARY 6.3.20. If Θ is a set of formulas, then the following are equivalent: (a)

Θ is linear,

(b)

Ł(Θ) is an MV-chain.

Proof. If Θ is inconsistent, then Ł(Θ) is a trivial MV-algebra, so our conclusion is obvious. If Θ is a consistent set, then the desired equivalence follows by Proposition 6.3.17 (b), Corollary 6.3.19 and Proposition 4.1.3. COROLLARY 6.3.21. If Θ is a set of formulas, then the following are equivalent: (a)

Θ is maximally consistent,

(b)

Ł(Θ) is a simple MV-algebra.

Proof. It follows by Proposition 6.3.15 (e), Corollary 6.3.19, Proposition 3.4.2 (the filter version) and Proposition 4.2.9.

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6.4

The semantics of Ł

DEFINITION 6.4.1 (Evaluation). Let A be an MV-algebra. An A-evaluation is a function e : Fm Š → A which satisfies the following conditions: (e1) e(ϕ → ψ) = e(ϕ) →A e(ψ), (e2) e(¬ϕ) = e(ϕ)∗ , where →A and ∗ are the implication and the negation operations in A. In the sequel, the operations of an MV-algebra A will be simply denoted ⊕, , ∗ , →, ∨, ∧, without any qualification. The reader can deduce from the context if a symbol operations is a logical one, or it denotes an MV-algebra operation. A function e : Fm Š → [0, 1] will be called an evaluation if it is a [0, 1]-evaluation. FACT 6.4.2. If A is an MV-algebra, then it suffices to define an A-evaluation on propositional variables. Given e : V → A, we can inductively define an evaluation e# : Fm Š → A as follows:  e(ϕ) if ϕ ∈ V ,   # # # e (ψ) → e (χ) if ϕ is ψ → χ, e (ϕ) =   # e (ψ)∗ if ϕ is ¬ψ. Thus, an A-evaluation is uniquely determined by the values of the propositional variables. In the following we will use the same notation for a function e : V → A and the corresponding A-evaluation e : Fm Š → A. DEFINITION 6.4.3 (Tautologies). If A is an MV-algebra, we say that a formula ϕ is an A-tautology if e(ϕ) = 1 for any A-evaluation e : Fm Š → A. We say that a formula ϕ is a tautology if it is a [0, 1]-tautology. We will denote by |=A ϕ the fact that ϕ is an A-tautology and by |= ϕ the fact that ϕ is a tautology. In the sequel, Taut A will be the set of all the A-tautologies and T aut will be the set of all the tautologies of Š. DEFINITION 6.4.4 (Semantic consequence). Let Θ be a set of formulas and ϕ a formula. For an MV-algebra A, we say that ϕ is an A-semantic consequence of Θ if e(Θ) = {e(ψ) | ψ ∈ Θ} = {1} implies e(ϕ) = 1, for any A-evaluation e : Fm Š → A. In this case we write Θ |=A ϕ. If ϕ is a [0, 1]semantic consequence of Θ, we will simply say that ϕ is a semantic consequence of Θ, and we write Θ |= ϕ. The set of all the A-semantic consequences of Θ will be denoted by Taut A (Θ) and, consequently, the set of all the semantic consequences of Θ will be Taut(Θ). One can see that A-tautologies (tautologies) coincide with A-semantic consequences (semantic consequences) of the empty set. PROPOSITION 6.4.5 (Soundness. Completeness w.r.t. MV-chains). If Θ is a set of formulas and ϕ is a formula, then the following are equivalent: (a)

Θ ` ϕ,

(b)

Θ |=A ϕ for any MV-algebra A,

(c) Θ |=A ϕ for any MV-chain A.

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Proof. (a) ⇒ (b) Let Θ ` ϕ and let ϕ1 , . . . , ϕn = ϕ be a Θ-proof for ϕ. For any MV-algebra A and for any A-evaluation e such that e(Θ) = {1}, we will prove that e(ϕi ) = 1 for any i ∈ {1, . . . , n} using mathematical induction. If i = 1, then it follows that ϕ1 ∈ Θ or ϕ1 is an axiom. If ϕ1 ∈ Θ it is obvious that e(ϕ1 ) = 1. If ϕ1 is one of the axioms (A2), (A3) or (A4), then e(ϕ1 ) = 1 from Proposition 2.3.5. If ϕ1 is (A1), then e(ϕ1 ) = 1 by Corollary 2.3.7 and Proposition 2.3.8 (e). Now, we suppose that e(ϕj ) = 1 for any j < i and we prove that e(ϕi ) = 1. If ϕi ∈ Θ or ϕi is an axiom, then e(ϕi ) = 1 as above. Otherwise, there are j, k < i such that ϕk is ϕj → ϕi . By induction hypothesis, we infer that 1 = e(ϕk ) = e(ϕj ) → e(ϕi ) = 1 → e(ϕi ). It follows that e(ϕi ) = 1. (b) ⇒ (c) Obvious. (c) ⇒ (a) We suppose that Θ 6` ϕ. By Proposition 6.3.9, it follows that there is a linear set of formulas Γ such that Θ ⊆ Γ and Γ 6` ϕ. We denote C = Ł(Γ), the Lindenbaum– Tarski algebra of Γ. By Corollary 6.3.20, C is an MV-chain. Let e : Fm Š → C defined by e(ψ) = [ψ]Γ , which obviously is a C-evaluation. Note that e(Γ) = {1}, so e(Θ) = {1} since Θ ⊆ Γ. By hypothesis, it follows that e(ϕ) = 1, which means that [ϕ]Γ = Theor (Γ). Thus, Γ ` ϕ, which is a contradiction. We proved that Θ ` ϕ. COROLLARY 6.4.6. If Θ is a set of formulas, then \ Theor (Θ) = {Taut A (Θ) | A is an MV-algebra}. Proof. By Proposition 6.4.5. COROLLARY 6.4.7. Theor ⊆ T aut. Proof. It follows by Corollary 6.4.6, when Θ = ∅ and A = [0, 1]. COROLLARY 6.4.8. The classical deduction theorem does not hold in Š. Proof. We recall that in classical (Boolean) logic the deduction theorem asserts that for any set of formulas Θ and for any two formulas ϕ and ψ, Θ ∪ {ϕ} ` ψ iff Θ ` ϕ → ψ. If Θ ` ϕ → ψ, then we also have Θ ∪ {ϕ} ` ψ by Theorem 6.1.7 (the deduction theorem in Š). We give an counter-example for the converse implication. By (14), we have ` ϕ → (ϕ → (ϕ ϕ)), so {ϕ} ` ϕ ϕ. If we suppose that ` ϕ → (ϕ ϕ) then, by Corollary 6.4.7, e(ϕ → (ϕ ϕ)) = 1 for any evaluation e : Fm Š → [0, 1]. It follows that 1 = e(ϕ)∗ ⊕ (e(ϕ) e(ϕ)) = e(ϕ)∗ ∨ e(ϕ). If ϕ is a propositional variable and we choose e(ϕ) = 1/2, then we get a contradiction. Thus, {ϕ} ` ϕ ϕ does not imply that ` ϕ → (ϕ ϕ). COROLLARY 6.4.9. The empty set ∅ is consistent. Proof. We suppose that the ∅ is inconsistent. By Lemma 6.3.2, there is a formula ϕ such that ` ϕ and ` ¬ϕ. If e : Fm Š → [0, 1] is an evaluation then, by Corollary 6.4.7, e(ϕ) = 1 and e(¬ϕ) = 1. Thus, we get 1 = e(ϕ) = e(¬ϕ) = e(ϕ)∗ = 0, which is a contradiction. We proved that ∅ is consistent.

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COROLLARY 6.4.10. For a nonempty set of formulas Θ the following properties are equivalent: (a)

Θ is consistent,

(b)

there is an evaluation e : Fm Š → [0, 1] such that e(Θ) = {1}.

Proof. (a) ⇒ (b) By Corollary 6.3.3, there is a maximally consistent set of formulas Γ such that Θ ⊆ Γ. From Corollary 6.3.21, we infer that Ł(Γ) is a simple MV-algebra, so it is isomorphic with a subalgebra of [0, 1] by Proposition 5.4.1. Thus, there is an injective homomorphism f : Ł(Γ) → [0, 1]. We define e : Fm Š → [0, 1] by e(ϕ) = f ([ϕ]Γ ). If ϕ ∈ Θ ⊆ Γ, then [ϕ]Γ = 1, so e(ϕ) = 1. Hence, e is the desired evaluation. (b) ⇒ (a) Let e be an evaluation such that e(Θ) = {1}. If Θ is inconsistent then, by Lemma 6.3.2, there is a formula ϕ such that Θ ` ϕ and Θ ` ¬ϕ. By Proposition 6.4.5, e(ϕ) = e(¬ϕ) = 1, so 0 = 1 in [0, 1], which is a contradiction. It follows that Θ is a consistent set of formulas. COROLLARY 6.4.11. If Θ is a set of formulas such that for any finite subset Γ ⊆ Θ there is an evaluation eΓ such that eΓ (Γ) = {1}, then there is an evaluation e such that e(Θ) = {1}. Proof. By hypothesis and Corollary 6.4.10, every finite subset of Θ is consistent. We suppose that there is no evaluation e such that e(Θ) = {1}. By Corollary 6.4.10, Θ is inconsistent. From Lemma 6.3.2, there is a formula ϕ such that Θ ` ϕ ¬ϕ. Using Lemma 6.1.2, there is a finite set of formulas Γ ⊆ Θ such that Γ ` ϕ ¬ϕ. Thus, Γ is also inconsistent, which is a contradiction. Recall that [0, 1]Q is the MV-algebra of the rational numbers from the unit interval [0, 1]. THEOREM 6.4.12 (Completeness). If Θ is a finite set of formulas and ϕ is a formula, then the following assertions are equivalent: (a)

Θ ` ϕ,

(b)

Θ |= ϕ,

(c) Θ |=Łn ϕ for any n ≥ 2, (d)

Θ |=[0,1]Q ϕ.

Proof. (a) ⇒ (b) Follows by Proposition 6.4.5. (b) ⇒ (c) Let n ≥ 2 and let e : Fm Š → Łn be an Łn -evaluation such that e(Θ) = {1}. Since Łn is a subalgebra of [0, 1], using (b), it follows that e(ϕ) = 1. (c) ⇒ (d) We suppose that Θ 6|=[0,1]Q ϕ, so there is a [0, 1]Q -evaluation e such that e(Θ) = {1} and e(ϕ) 6= 1. As we already noticed, e is uniquely determined by the values e(v), where v is a propositional variable. Let v1 , . . . , vn be all the propositional variables that appear in ϕ or in the formulas from Θ and let e(vi ) = mi /di ∈ [0, 1]Q for any i ∈ {1, . . . , n}. Hence, mi and di are natural numbers such that di 6= 0 for any i ∈ {1, . . . , n}. If we consider d the least common multiple of d1 , . . . , dn , then mi /di = pi /d and 0 ≤ pi ≤ d for any i ∈ {1, . . . , n}. It follows that e(vi ) ∈ {0, 1/d, . . . , (d − 1)/d, 1} = Ld+1 for any i ∈ I. We consider an Łd+1 -evaluation

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e0 : Fm Š → Ld+1 with the property that e0 (vi ) = e(vi ) = pi /d for any i ∈ {1, . . . , n}. Since, for any formula ψ, the value e0 (ψ) is determined by the values of the propositional variables that appear in ψ, we infer that e0 (ϕ) = e(ϕ) 6= 1 and e0 (Θ) = e(Θ) = {1}. Thus, Θ 6|=Łd+1 ϕ, so we get a contradiction of (c). It follows that Θ |=[0,1]Q ϕ. (d) ⇒ (a) Let us suppose that Θ 6` ϕ. By Proposition 6.4.5, there is an MV-chain A and there is an A-evaluation e such that e(Θ) = {1} and e(ϕ) 6= 1. By Corollary 5.3.1, we can safely assume that A is divisible. Assume Θ = {θ1 , . . . , θk } and let v1 , . . . , vn be all the propositional variables that appear in ϕ or in a formula from Θ. We define the following sentence in the first order language of MV-algebras: σ is (∀v1 ) . . . (∀vn )((θ1 = 1 ∧ · · · ∧ θk = 1) → ϕ = 1). If we consider the formulas ϕ = 1(v1 , . . . , vn ) and θi = 1(v1 , . . . , vn ) for any i ∈ {1, . . . , k}, then the previous assumption asserts that A |= θi = 1[a1 , . . . , an ] for any i ∈ {1, . . . , k} and A 6|= ϕ = 1[a1 , . . . , an ], where aj = e(vj ) for any j ∈ {1, . . . , n}. It follows that A 6|= σ. By Theorem 5.3.6, we infer that [0, 1]Q 6|= σ, so there are rational numbers r1 , . . . , rn in [0, 1] such that for any i ∈ {1, . . . , k} we have: [0, 1]Q |= θi = 1[r1 , . . . , rn ] and

[0, 1]Q 6|= ϕ = 1[r1 , . . . , rn ].

Let e0 : Fm Š → [0, 1]Q be a [0, 1]Q -evaluation such that e0 (vj ) = rj for any j ∈ {1, . . . , n}. We get e0 (θi ) = 1 for any i ∈ {1, . . . , k} and e0 (ϕ) 6= 1, which is a contradiction of (d). Hence, Θ ` ϕ. COROLLARY 6.4.13. If Θ is a finite set of formulas, then Theor (Θ) = Taut(Θ). In particular, Theor = Taut. Proof. By Theorem 6.4.12. In the sequel we will provide a characterization of those sets of formulas Θ with the property that the syntactic consequences and the semantic consequences of Θ coincide. The following example shows that this property does not hold in general. EXAMPLE 6.4.14 ([80]). Let v and u be two propositional variables and Θ = {nv → u | n ≥ 1} ∪ {(¬v) → u}. We will prove that Θ |= u, but Θ 6` u. Let e : Fm Š → [0, 1] be an evaluation such that e(Θ) = {1}. Thus e(nv → u) = e(¬v → u) = 1 for any n ≥ 1, so e(v)∗ ≤ e(u) and ne(v) ≤ e(u) for any n ≥ 1. If e(v) = 0, then e(v)∗ = 1 and we get e(u) = 1. If e(v) > 0, since [0, 1] is a simple MValgebra, there is n ≥ 1 such that ne(v) = 1, so e(u) = 1. We proved that e(Θ) = {1} implies e(u) = 1, which means that Θ |= u. We suppose that Θ ` u. By Lemma 6.1.2, there is a finite set Γ ⊆ Θ such that Γ ` u. From Theorem 6.4.12 we infer that Γ |= u. We will define an evaluation e such that e(Γ) = {1}, but e(u) 6= 1, so we will get a contradiction. Note that Γ is not empty, since u is a propositional variable. We define M = {n | nv → u ∈ Γ}. Let m = max{n | n ∈ M } if M 6= ∅ and m = 1, otherwise. We consider e an evaluation

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such that e(v) = 1/(m + 1) and e(u) = m/(m + 1). If nv → u ∈ Γ then, since n ≤ m, it follows that e(nv → u) = ne(v) → e(u) = (n/(m + 1)) → (m/(m + 1)) = 1. If ¬v → u ∈ Γ, then e(¬v → u) = e(v)∗ → e(u) = (1 − (1/(m + 1))) → (m/(m + 1)) = (m/(m + 1)) → (m/(m + 1)) = 1. Thus, e(Γ) = {1}, but e(u) = m/(m + 1) 6= 1. Example 6.4.14 also shows that the semantical compactness does not hold in Š. This means that Θ |= ϕ does not imply that there is a finite subset Γ ⊆ Θ such that Γ |= ϕ. For a geometric representation of the same phenomenon we refer to [61]. We also note that in classical (Boolean) logic, the semantical compactness is equivalent to the previous Corollary 6.4.11. LEMMA 6.4.15. If Θ is an arbitrary set of formulas, then for any MV-algebra A and for any A-evaluation e : Fm Š → A such that e(Θ) = {1}, there is a unique MValgebra homomorphism fe : Ł(Θ) → A with fe ([ϕ]Θ ) = e(ϕ) for any ϕ ∈ Fm Š . Proof. Let us define fe : Ł(Θ) → A by fe ([ϕ]Θ ) = e(ϕ) for any ϕ ∈ Fm Š . We have to prove that fe is well defined. If ϕ and ψ are formulas such that [ϕ]Θ = [ψ]Θ , then Θ ` ϕ → ψ and Θ ` ψ → ϕ. By Proposition 6.4.5 and our hypothesis, it follows that e(ϕ → ψ) = e(ψ → ϕ) = 1, so e(ϕ) = e(ψ). Thus, fe is well defined. The fact that fe is a homomorphism and its uniqueness are straightforward. COROLLARY 6.4.16. Let A be an MV-algebra, Θ a set of formulas and ϕ a formula. The following are equivalent: (a) (b)

Θ |=A ϕ, f ([ϕ]Θ ) = 1 for any MV-algebra homomorphism f : Ł(Θ) → A.

Proof. (a) ⇒ (b) If f : Ł(Θ) → A is an MV-algebra homomorphism, then we define an A-evaluation by e(ψ) = f ([ψ]Θ ). For any ψ ∈ Θ we get e(ψ) = 1, so e(ϕ) = 1 by hypothesis. Thus, f ([ϕ]Θ ) = 1. (b) ⇒ (a) Let e : Fm Š → A be an A-evaluation such that e(Θ) = {1} and let fe be the homomorphism uniquely determined by e from Lemma 6.4.15. By hypothesis, it follows that fe ([ϕ]Θ ) = 1, so e(ϕ) = 1. LEMMA 6.4.17. If v1 6= v2 are propositional variables, then [v1 ] 6= [v2 ]. Proof. Let A be an MV-algebra and e : Fm Š → A an A-evaluation such that e(v1 ) 6= e(v2 ). By Lemma 6.4.15 there is a unique MV-algebra homomorphism fe : Ł → A with fe ([v1 ]) = e(v1 ) 6= e(v2 ) = fe ([v2 ]). If we assume that [v1 ] = [v2 ] then, by completeness, |=A v1 → v2 and |=A v2 → v1 . Hence, by Corollary 6.4.16, we get fe ([v1 ]) = fe ([v2 ]), which contradicts our hypothesis. We proved that [v1 ] = [v2 ]. LEMMA 6.4.18. The following are equivalent for any MV-algebra A and a ∈ A: (a) (b)

a ∈ U for any ultrafilter U of A, f (a) = 1 for any MV-algebra homomorphism f : A → [0, 1].

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Proof. (a) ⇒ (b) If f : A → [0, 1] is an MV-algebra homomorphism, then U = {x ∈ A | f (x) = 1} is a filter. By Theorem 3.2.8, A/U is isomorphic with f (A). Since f (A) is a subalgebra of [0, 1], it is a simple MV-algebra, so U is an ultrafilter. By hypothesis, a ∈ U , so f (a) = 1. (b) ⇒ (a) If U is an ultrafilter of A, then A/U is a simple MV-algebra, so there is an injective homomorphism f 0 : A/U → [0, 1]. We define f : A → [0, 1] by f (x) = f 0 ([x]U ). It is obvious that f is an MV-algebra homomorphism, so f (a) = 1 by hypothesis. Thus, f 0 ([a]U ) = 1, so [a]U = 1 since f 0 is injective. We proved that a ∈ U . THEOREM 6.4.19. If Θ is a set of formulas, then the following are equivalent: (a)

Theor (Θ) = Taut(Θ),

Ł(Θ), the Lindenbaum–Tarski algebra of Θ, is semisimple, T (c) [Theor (Θ)] = {U ⊆ Ł(Θ) | [Theor (Θ)] ⊆ U , U ultrafilter}.

(b)

Proof. (a) ⇔ (b) By Corollary 6.4.16 and Lemma 6.4.18, we get [ϕ]Θ ∈ U if and only if ϕ ∈ Taut(Θ) for any ultrafilter U of Ł(Θ). By Lemma 6.2.2 (a) it follows that, if Taut(Θ) = Theor (Θ), the intersection of all the ultrafilters of Ł(Θ) is {Theor (Θ)}. Thus, in the MV-algebra Ł(Θ), the intersection of all the ultrafilters contains only the last element of the algebra, so Ł(Θ) is semisimple. Conversely, if Ł(Θ) is semisimple, then the intersection of all the ultrafilters is Theor (Θ). Thus, if ϕ ∈ Taut(Θ), then [ϕ]Θ = Theor (Θ), which means that ϕ ∈ Theor (Θ). We proved that Taut(Θ) ⊆ Theor (Θ). Since Theor (Θ) ⊆ Taut(Θ) by Corollary 6.4.6, (a) is proved. (b) ⇔ (c) Follows by Corollary 6.3.19 and the dual of Proposition 3.2.7.

7

Varieties of MV-algebras

In this section we describe equational classes, i.e. varieties of MV-algebras, that is classes V of MV-algebras for which exists a set E of MV-equations such that an MValgebra A is in V if and only if every equation from E holds in A. We recall that a variety of algebras is a class of algebras which is closed under subalgebras, homomorphic images and direct products. The class of all MV-algebras, denoted by MV, is obviously an equational class. Let {Ai }i∈I be a family of MV-algebras. Then by V({Ai }i∈I ) we denote the variety generated by {Ai }i∈I , i.e., the smallest variety of MV-algebras containing {Ai }i∈I . The algebras Ai will be called generators of V({Ai }i∈I ). Also we write V(A) to denote the variety generated by {A} or V(A1 , . . . , An ) to denote the variety generated by {A1 , . . . , An }, for any positive integer n. When V(A1 , . . . , An ) cannot be generated by a proper subset of {A1 , . . . , An }, we say that {A1 , . . . , An } is an irreducible set of generators. In the sequel we write just V(A1 , . . . , An ) to denote a variety generated by an irreducible set of generators {A1 , . . . , An }. In the present section we mainly show that every proper subvariety of the variety MV of all MV-algebras is generated by a finite set of MV-chains and that it is axiomatizable by a finite set of equations in a single variable. Throughout this section we write rank(A) = n whenever A is an MV-algebra of rank n.

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7.1

Komori’s Theorem

Let A be an MV-algebra and τ, σ MV-terms in the variables x1 , . . . , xn such that for all a1 , . . . , an ∈ A: τA (a1 , . . . , an ) = σA (a1 , . . . , an ). In this case we say that the MV-algebra A satisfies the MV-equation τ = σ, or that the MV-equation τ = σ holds in A, and write A |= τ = σ. We simply write τ = σ to refer to the pair hτ, σi. The following proposition is a standard result of universal algebra. PROPOSITION 7.1.1. Let A be an MV-algebra which is a subdirect product of the MV-algebras {Ai }i ∈ I and σ, τ MV-terms in the variables x1 , . . . , xn . Then the following are equivalent: (a)

A |= τ = σ,

(b)

Ai |= τ = σ for every i ∈ I.

THEOREM 7.1.2. Every variety of MV-algebras is generated by the collection of its MV-chains. Proof. By Theorem 4.1.4 every MV-algebra A is subalgebra of the product of MVchains Ai , i ∈ I, where each Ai is a quotient of A by a prime ideal of A. Furthermore, by Proposition 7.1.1 any MV-equation τ = σ is valid in A iff it is valid in Ai , for every i ∈ I. The proof is indeed complete. PROPOSITION 7.1.3. For any infinite subalgebra A of [0, 1], V(A) = MV. Proof. By Corollary 4.1.12, A is dense in [0, 1]. Since for any MV-term τ , the function τ[0,1] is continuous, an equation τ = σ is valid in A iff it is valid in [0, 1]. PROPOSITION 7.1.4. If 2 < n0 < n1 < · · · is an infinite sequence of natural numbers, then V({Łni | i = 0, 1, . . . }) = MV. S Proof. Let S = ( Lni )i=0,1,... and assume that the MV-equation σ = τ , in k variables, is satisfied in each Łni . Let u1 , . . . , uk ∈ [0, 1]. Since S is infinite, for every ui ∈ {u1 , . . . , uk } there is an increasing infinite sequence of elements of S, yi1 < · · · < yim · · · such that yim < ui for every positive integer m. From the continuity of MV-terms on [0, 1]k we get that σ = τ holds in [0, 1] too. Vice versa, trivially, every MV-equation valid in [0, 1] is valid in each Łni , then the proposition is proved. PROPOSITION 7.1.5. Let W be a proper subvariety of MV. Then there is an integer m ≥ 1 such that for each MV-chain A of W rank(A) ≤ m. Proof. Let A be an MV-chain and A ∈ W. Then A/Rad (A) ∈ W. By Proposition 7.1.3 A/Rad (A) has to be finite, i.e., A is of finite rank. By Proposition 7.1.4 the set of finite chains from W has to be finite, so the set of ranks of chains from W is finite too. This proves the proposition. For any integer n ≥ 1 we set Hn = Γ(Z ×lex Z, hn, 1i). LEMMA 7.1.6. For each integer n ≥ 1 and each nonsimple MV-chain A of rank n, Hn ∈ V(A).

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Proof. By Proposition 7.1.5 we assume A ' Γ(Z ×lex G, hn, gi) for some totally ordered Abelian group G and 0 < g ∈ G. The mapping h : Hn → Γ(Z ×lex G, hn, gi) defined by h(hj, mi) = hj, mgi for each hj, mi is an injective MV-homomorphism and h(hn, 1i) = hn, gi. Hence it follows that Hn ∈ V(A). Note that Łn+1 ' Γ(Z, n). In the rest of this section, Łn+1 is to be intended as Γ(Z, n), instead of the subalgebra of [0, 1] with universe {0, 1/n, . . . , (n − 1)/n, 1}. For any integer n ≥ 1 we set K n+1 = Γ(Z ×lex Z, hn, 0i). LEMMA 7.1.7. Let A be an MV-algebra and m, n positive integers. If A contains an isomorphic image of Łn+1 and an isomorphic image of K m+1 , then A contains an isomorphic image of K n+1 . Proof. Let f : Łn+1 → A and Let g : K m+1 → A be embeddings. Set T = f (Ln+1 ) ∪ g(hRad (K m+1 )i, W = {f (k/n) (g(0, r))∗ }00 , V = {f (k/n) ⊕ g(0, r)}00 , S = T ∪ W ∪ V. Let us prove that S is a totally ordered subset of A. Indeed, it is easy to see that T is totally ordered. So we show now that W is totally ordered. Let a, b ∈ W such that a = f (k/n) (g(0, r))∗

and

b = f (h/n) (g(0, s))∗.

If h = k, trivially a and b are comparable. If k < h, by Lemma 3.5.13, b = (f (k/n) ((g(0, s))∗ ) ⊕ f ((h − k)/n). For r = s, b = a ⊕ f ((h − k)/n) > a. For r < s, by Lemma 3.5.11 (a) and Lemma 3.5.12, b = (f (k/n) (g(0, r))∗ ) ⊕ g(0, s) > a. For r > s, b > f (k/n) (g(0, r))∗ ) = a. Hence, W is a totally ordered set. Now, let a, b ∈ V and a = f (k/n) ⊕ g(0, r)

b = f (h/n) ⊕ g(0, s).

Then, as above, if k < h we have: b > f (k/n) ⊕ f ((h − k)/n) > a. If h = k, then a < b if and only if r < s, and a > b if and only if r > s. Hence V is totally ordered. To show that S is totally ordered we distinguish the following cases: (1)

a ∈ T and b ∈ W ,

(2)

a ∈ T and b ∈ V ,

(3)

a ∈ W and b ∈ V .

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Assume (1) and b = f (k/n) (g(0, r))∗ ). For a = f (k/n), if k = h, trivially we have b < a. If k > h, then a = f (k/n) ⊕ f ((h − k)/n) > b. If k < h, then by Lemma 3.5.13: b = (f (k/n) ⊕ f ((h − k)/n)) (g(0, r))∗ ) = = f (k/n) ⊕ (f ((h − k)/n)) g(0, r))∗ ) > a. For a = g(0, s) and b = f (k/n) (g(0, r))∗ ), by Lemma 3.5.11 (c), a < b. For a = (g(0, s))∗ , a∗ = g(0, s) < (f (k/n))∗ ⊕ g(0, r) = b∗ . Hence b < a. Assume (2) and b = f (k/n) g(0, r). For a = f (k/n), if k ≤ h, then a ≤ b. If k > h, then by Lemma 3.5.11 (a), a > b. For a = g(0, s), by Lemma 3.5.11 (a), a < b. For a = (g(0, s))∗ , a∗ = g(0, s) < (f (k/n))∗ (g(0, r))∗ = b∗ . Hence b < a. Assume (3) a = f (k/n) (g(0, s))∗ and b = f (k/n) ⊕ g(0, r). If k ≤ h, then a < b. If k > h, by Lemma 3.5.11 (a) and (g) a = (f (k/n) (g(0, s))∗ ) ⊕ f ((h − k)/n) > > ((f (k/n) (g(0, s))∗ ) ⊕ g(0, s)) ⊕ g(0, r) = = (f (k/n) ∨ g(0, s)) ⊕ g(0, r) = b. Hence, we proved that S is a totally ordered subset of A. Let us prove, now, that S is a subalgebra of A. It is clear that 0 = f (0) ∈ S, and 1 = f (1) ∈ S. To prove that S is closed under the operation ∗ it is just a matter of checking. To show that S is closed under ⊕ operation we limit ourselves to analyze the cases listed below, the remaining cases are trivial. We assume that x, y ∈ S. Case 1. x = f (k/n) ∈ T and y = (g(0, r))∗ ∈ T Then, by Lemma 3.5.11 (a), x ⊕ y = 1. Case 2. x = f (k/n) (g(0, r))∗ ∈ W and y = f (h/n) (g(0, s))∗ ∈ W If h + k = n then, by Lemma 3.5.11 (e), x ⊕ y = (g(0, r + s))∗ ∈ T . If h + k ≥ n + 1, then x ⊕ y ≥ (f (k/n) (g(0, r))∗ ) ⊕ ((f (k/n)∗ ⊕ (f (1/n)) (g(0, s))∗ ) ≥ ≥ (f (k/n) (g(0, r))∗ ) ⊕ ((f (k/n)∗ ⊕ g(0, r) ⊕ g(0, s)) g(0, r))∗ ) = = f (k/n) ⊕ f (k/n)∗ = 1. Case 3. x = f (k/n) ∈ T and y = f (h/n) (g(0, r))∗ ∈ W If h + k = n, then, by Lemma 3.5.11 (e) x ⊕ y = (g(0, r))∗ ∈ T . If h + k < n, then by Lemma 3.5.13 x ⊕ y ∈ W . If h + k ≥ n + 1, then by Lemma 3.5.11 (a) x ⊕ y = 1. Case 4. x = f (k/n) (g(0, s))∗ ∈ W and y = f (h/n) ⊕ g(0, r) ∈ V If h + k ≥ n, as above, by Lemma 3.5.11 (a) x ⊕ y = 1. If h + k < n, then by Lemma 3.5.13, if r > s, x ⊕ y = f ((h + k)/n) (g(0, r − s))∗ ∈ V , if r = s, x ⊕ y = f ((h + k)/n) ∈ T , if r < s, x ⊕ y = f ((h + k)/n) (g(0, s − r))∗ ∈ W .

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Hence, we proved that S is a subalgebra of A. Finally, we provide the desired isomorphism between S and K n+1 checking, by a direct inspection, that the mapping ψ defined below is an isomorphism: ψ(x) = hf −1 (x), 0i ψ(x) = g

−1

(x)

if x ∈ f (Ln+1 ), if x ∈ g(hRad (K m+1 )i ,

ψ(x) = hk/n, −ri

if x ∈ W and x = f (k/n) (g(0, r))∗ ,

ψ(x) = hk/n, ri

if x ∈ V and x = f (k/n) ⊕ (g(0, r))∗ .



LEMMA 7.1.8. Let A be an MV-algebra and m, n positive integers. If A contains an isomorphic image of K n+1 and an isomorphic image of K m+1 , then A contains an isomorphic image of K q+1 , where q = l.c.m.(m, n). Proof. Let f : K n+1 → A and g : K m+1 → A be isomorphisms. Then the restrictions of f and g to Ln+1 and to Lm+1 , respectively, are isomorphisms. It follows that A contains a copy of Łq+1 . The thesis follows by Lemma 7.1.7. PROPOSITION 7.1.9. Let G be an Abelian `-group, 0 < b ∈ G. Then for each n ≥ 1, Γ(Z ×lex G, hn, bi) ∈ V(K n+1 ). Proof. Let G be an Abelian `-group, 0 < b ∈ G and n > 0. For each integer n > 0, and 0 < b ∈ G, Γ(Z ×lex G, hn, bi) can be embedded into Γ(Z ×lex G, hn, 0i). Indeed, such an embedding can be obtained via the map ϕ defined as: ϕ(m, y) = (m, ny − mb). Hence we only need to show that Γ(Z ×lex G, hn, 0i) ∈ V(K n+1 ). By assumption, G is an Abelian `-group. There exists a subgroup K of the Abelian `-group ZI , for some index set I and a group `-homomorphism f such that f (K) = G. Via the functor ∆, we have: (j)

∆(f )(∆(K)) = ∆(G),

i.e., ∆(G) is a perfect MV-algebra which is a homomorphic image of ∆(K), and ∆(K) ⊆ ∆(ZI ).

(jj)

We define a homomorphism h as follows: h(0, a) = h0, f (a)i for every a ∈ K+ , h(m, a) = hm, f (a)i for every 0 < m < n and a ∈ K, h(n, a) = hn, f (a)i for every a ∈ K− . Note that (jjj)

h(Γ(Z ×lex K, hn, 0i)) = Γ(Z ×lex G, hn, 0i).

Hence we proved that Γ(Z ×lex G, hn, gi) ∈ V(K n+1 ). LEMMA 7.1.10. For any integer n ≥ 1, K n+1 ∈ V(Hn ).

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Proof. The Lemma is proved once we will show that if an MV-equation holds in Hn it has to be valid in K n+1 too. By Proposition 2.6.1 (d2), we can assume that the equation has the form π(x1 , . . . , xk ) = 0, for some MV-term π in the variables x1 , . . . , xk . Assume that the equation π(x1 , . . . , xk ) = 0 does not hold in K n+1 , for some MVterm π. Then there are elements c1 , . . . , ck ∈ Kn+1 such that πKn+1 (c1 , . . . , ck ) ≥ h0, 1i.

(1.1)

For each integer m ≥ 1, let fm : Z×lex Z → Z×lex Z be the `-group homomorphism defined by fm (v, w) = hv, mwi f or all hv, wi ∈ Z. For an MV-term σ in the variables x1 , . . . , xk let g(σ) be the number of symbols ∗ and ⊕ occurring in σ. By induction on g(σ), for each k-tuple hb1 , . . . , bk i ∈ Kn+1 ⊆ Hn , we will prove the following system of inequalities: fm (σKn+1 (b1 , . . . , bk )) − (0, g(σ)) ≤ σHn (fm (b1 ), . . . , fm (bk )),

(1.2)

σHn (fm (b1 ), . . . , fm (bk )) ≤ fm (σKn+1 (b1 , . . . , bk )) + (0, g(σ)).

Indeed, if g(σ) = 0, then σ = xi , i.e., σ is a single variable or σ = 0, the constant 0. The latter case is trivial, so we have to show that fm (σKn+1 (bi )) ≤ σHn (fm (bi ))). Let bi = hv, wi ∈ Kn+1 ⊆ Hn , then σKn+1 (bi ) = bi and σHn (fm (bi ))) = fm (bi ). Hence fm (σKn+1 (b1 , . . . , bk )) = fm (bi ) = (v, mw) = σHn (fm (b1 ), . . . , fm (bk )). Thus the desired inequalities are proved in the case g(σ) = 0. We proceed by induction on the number of symbols ∗ and ⊕ occurring in MV-terms. Suppose that for some integer d > 0 (1.2) holds for all MV-terms ξ in the variables x1 , . . . , xk such that g(ξ) < d. Let σ(x1 , . . . , xk ) be an MV-term such that g(σ) = d. Only the following cases occur. Case 1.

There is an MV-term τ (x1 , . . . , xk ) such that σ = τ ∗ .

Case 2.

There are MV-terms µ(x1 , . . . , xk ), ν(x1 , . . . , xk ) such that σ = µ ⊕ ν.

We can safely restrict ourselves to considering MV-terms in one variable. Assume the Case 1 holds. Let us prove that: σHn (fm (b)) ≤ fm (σKn+1 (b)) + h0, g(σ)i.

(1)

Indeed, we have g(σ) = 1 + g(τ ), (i)

σHn (fm (b)) = hn − 1, 1i − τHn (fm (b))

and, by induction hypothesis (ii)

τHn (fm (b)) ≤ fm (τKn+1 (b)) + h0, g(τ )i

(iii)

fm (τKn+1 (b)) − h0, g(τ )i ≤ τHn (fm (b)).

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By (i) and (ii) we get: σHn (fm (b)) ≥ hn − 1, 1i − (fm (τKn+1 (b)) − h0, g(τ )i = = hn − 1, 1i − fm (τKn+1 (b)) + h0, 1i − h0, g(σ)i ≥ ≥ fm (σKn+1 (b)) − h0, g(σ)i. Similarly, by (i) and (iii): σHn (fm (b)) ≤ hn − 1, 1i − fm (τKn+1 (b)) + h0, g(τ i) = = hn − 1, 0i − fm (τKn+1 (b)) + h0, 1i + h0, g(τ )i = = fm (σKn+1 (b)) + h0, g(σ)i. Hence σ satisfies (1). In Case 2, we have g(σ) = g(µ) + g(ν) + 1, (iv) σHn (fm (b))

= hn − 1, 1i ∧ (µHn (fm (b)) + νHn (fm (b)))

and by induction hypothesis: σHn (fm (b))

≤

hn − 1, 1i ∧ (fm (µKn+1 (b))+ +fm (νKn+1 (b)) + h0, g(µ)i + h0, g(ν)i),

(vi) σHn (fm (b))

≥

hn − 1, 1i ∧ (fm (µKn+1 (b))+ +fm (νKn+1 (b)) − h0, g(ν)i − h0, g(µ)i).

(v)

Thus, by (iv) and (v) we get σHn (fm (b)) ≤ hn − 1, 1i ∧ (fm (µKn+1 (b)) + fm (νKn+1 (b))) + h0, g(µ) + g(ν)i ≤ ≤ hn − 1, 0i ∧ (fm (µKn+1 (b)) + fm (νKn+1 (b))) + h0, g(µ) + g(ν) + 1i = fm (σKn+1 (b)) + h0, g(σ)i. Similarly, by (iv) and (vi) we get σHn (fm (b)) ≥ fm (σKn+1 (b)) − h0, g(σ)i. Thus, we showed that (1) holds. Hence, by (1.1) and (1) we obtain (2) h0, m − g(π)i ≤ fm (πKn+1(b1 , . . . , bk ))−h0, g(π)i ≤ πHn(fm (b1 ), . . . , fm (bk )). Hence, if m > g(π), then πHn (fm (b1 ), . . . , fm (bk )) > h0, 1i and Hn does not satisfy the equation π(x1 , . . . , xk ) = 0. We conclude that K n+1 must satisfy all equations that are satisfied by Hn . THEOREM 7.1.11. For each integer n ≥ 1 and each non simple MV-chain A of rank n, V(A) = V(K n+1 ). Proof. From Theorem 5.5.6 and Lemmas 3.5.11 and 3.5.12. For each n = 1, 2, . . . let T(n) denote the subvariety of MV defined by the identity: Eq(T )

nx = (n + 1)x.

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LEMMA 7.1.12. Let m be an integer such that 1 ≤ m ≤ n. Then Łm+1 ∈ T(n). Proof. Trivial checking. LEMMA 7.1.13. Let m be an integer such that 1 ≤ n < m. Then Łm+1 ∈ / T(n). Proof. It is easy to check that for m > 1, the atom 1/m ∈ T(n), does not satisfy Eq(T ). LEMMA 7.1.14. Let A be an infinite simple MV-chain. Then A ∈ / T(n). Proof. Suppose A ∈ T(n). Then, by Proposition 7.1.3, A generates all the variety MV. By Lemma 7.1.13, T(n) is proper, absurd. Hence A ∈ / T(n). LEMMA 7.1.15. Let A be an infinite non simple MV-chain. Then A ∈ / T(n). Proof. Since A is not simple, then Rad (A) 6= {0}. Let x ∈ Rad (A) \ {0}, then it is easy to check that x does not satisfy Eq(T ). PROPOSITION 7.1.16. T(n) = V(Ł2 , . . . , Łn+1 ). Proof. By Lemmas 7.1.10, 7.1.12, and 7.1.13 and Theorems 7.1.2, 7.1.11. THEOREM 7.1.17. Let A be an MV-algebra and n ≥ 1 an integer. Then A ∈ T(n) if and only if A is subdirect product of Łk+1 algebras with 1 ≤ k ≤ n. Proof. By Proposition 4.1.3 and Lemma 7.1.14. For each n = 1, 2, . . . let W(n) denote the subvariety of MV defined by the identity: Eq(E)

((n + 1)xn )2 = 2xn+1 .

LEMMA 7.1.18. If A is an infinite atomless subalgebra of [0, 1], then A ∈ / W(n). Proof. We will show that there exists an element a ∈ A which does not satisfy Eq(E). Let q1 = (1 + 2(n2 − 1))/(2n(n + 1)) and q2 = n/(n + 1), then we get q1 < q2 and q1 , q2 ∈ Z ∩ [0, 1]. Since, by Proposition 4.1.9, A is dense in [0, 1], then there is a ∈ A such that q1 < a < q2 . It is easy to check that the element a ∈ A does not satisfy Eq(E). Hence A ∈ / W(n). LEMMA 7.1.19. For every integer 1 ≤ m ≤ n, K m+1 ∈ W(n). Proof. For n = 1 the result follows from a direct verification. In the case n > 1, if y ∈ (Rad (K m+1 ))∗ , it is easy to check that ((n + 1)y n )2 = 2y n+1 = 1. Let y = hm − 1, hi ∈ Km+1 , with h ∈ Z+ . Then ((n + 1)y n )2 = 0. On the other hand y n+1 = 0, and Eq(E) holds for every element of the set B = (Rad (K m+1 ))∗ ∪ {hm − 1, hi | h ∈ Z+ }. Finally, if y ∈ Km+1 \ B, then we can write y < z, for some z ∈ {hm − 1, hi | h ∈ Z+ } ⊆ Km+1 . By monotonicity we get ((n + 1)y n )2 = 2y n+1 = 0. LEMMA 7.1.20. Let 1 ≤ n < m. Then Łm+1 ∈ / W(n).

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Proof. Let 1 ≤ n < m be integers with m < 2(n + 1). Then the MV-algebra Łm does not satisfy the identity ((n + 1)y n )2 = 2y n+1 . Indeed, let y = m − 1 ∈ Lm+1 . Then y is the co-atom of Łm+1 and ((n + 1)y n )2 = 1. On the other hand, y n+1 = m − n − 1, whence, from m < 2(n + 1) and 2 < ord (y n+1 ) we get 2y n+1 < 1. Assume, now, 2(n + 1) ≤ m. Let the integers q and r be given by m = 2q(n + 1) + r, 0 < q, 0 ≤ r < 2(n + 1), and y = m − (q + 1) ∈ Lm+1 . Then it is easy to see that m − n(q + 1) ≥ m/2 + q − n ≥ q + 1 > 0, y n = m − n(q + 1) > 0, and (n + 1)y n = m(n + 1)(q + 1). By direct inspection we have: m(n + 1) − n((n + 1)q + (n + 1)) ≥ m + n(m/2 − (n + 1)). Thus, by hypothesis 2(n+1) ≤ m we obtain m(n+1)−n((n+1)q+(n+1)) ≥ m. Hence (n + 1)y n = ((n + 1)y n )2 = 1. Now we are going to prove that 2y n+1 6= 1. If y n+1 = 0, we are done. Suppose y n+1 > 0. Then, y n+1 = m − (n + 1)(q + 1). But, m − q(n + 1) − (n + 1) = m + r/2 − m/2 − (n + 1) < m/2. In conclusion, 2y n+1 < 1, whence ((n + 1)y n )2 6= 2y n+1 , as required. LEMMA 7.1.21. Let 1 ≤ n be an integer and A an MV-chain. Then A satisfies Eq(E) if and only if rank(A) ≤ n. Proof. By Lemma 7.1.18, Eq(E) holds in the algebra K n+1 , for 1 ≤ m ≤ n. Also, by Lemma 7.1.10, Eq(E) holds in all MV-chains of rank m ≤ n. By Lemma 7.1.18, for n < m, the MV-chain Łm+1 does not satisfy Eq(E). Hence, every MV-chain of rank m > n cannot satisfy Eq(E). THEOREM 7.1.22. W(n) = V(K 2 , . . . , K n+1 ). Proof. By Lemma 7.1.19 and Propositions 7.1.9, 4.1.3, and 7.1.1. THEOREM 7.1.23 (Komori’s theorem). Let W be a proper variety of MV-algebras. Then there are two finite sets I and J of integers greater than 2 such that I ∪ J 6= ∅ and W = V({Łi+1 }i∈I , {K i+1 }i∈J ). Proof. By Proposition 7.1.5 and Lemmas 7.1.15 and 7.1.20 we know that a variety of MV-algebras is proper if and only if it is generated by a set of MV-chains of bounded rank. An application of Proposition 7.1.9 then yields the theorem. 7.2

Equational characterization of varieties of MV-algebras For any i ∈ Z+ = {1, 2, . . . } we set δ(i) = {n ∈ Z | 1 ≤ n is a divisor of i}.

For J nonempty finite subset of Z+ and i = {1, 2, . . . } we let [ ∆(i, J) = {d ∈ δ(i) \ δ(j)}. j∈J

If J = ∅ we define ∆(i, J) = δ(i). For each integer n ≥ 3, let W(n, p) denote the variety defined by the identity: Eq(2p)

(pxp−1 )n+1 = (n + 1)xp

with 1 < p < n.

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THEOREM 7.2.1. For all n ≥ 3, and 1 < p < n, K n+1 ∈ W(n, p). Proof. Assume y ∈ Kn+1 and y p−1 = 0. Then y p = 0 and y satisfies Eq(2p). Assume now y = hs, zi ∈ Kn+1 and 0 < y p−1 . If y p = 0, then s < (p − 1)n/p. Hence, ps(p−1)−pn(p−2) < n. It follows that py p−1 = (ps(p−1)−pn(p−2), pz(p−1)) ∈ / (Rad (K n+1 ))∗ , and hence (py p−1 )n+1 = 0. If, on the other hand, y p > 0, then s > (p − 1)n/p, whence py p−1 = 1. We conclude that, for all y ∈ Kn+1 , Eq(2p) holds. THEOREM 7.2.2. Let n ≥ 3 and 1 < p < n. Then the following are equivalent for all 1 ≤ q < n: (i) Łq+1 ∈ W(n, p); (ii)

p does not divide q.

Proof. Assume that Łq+1 ∈ W(n, p) and p = kq for some k ∈ Z+ . Consider y = k(p − 1) ∈ Lq+1 \ {1}. Then we have py p−1 = pk(p − 1)2 − pq(p − 2) = q, and y p = pk(p − 1) − q(p − 1) = 0, a contradiction. Conversely, assume p does not divide q. and pick an arbitrary element y ∈ Lq+1 . If y p−1 = 0, then y q = 0 and Eq(2p) is satisfied. If, on the other hand, 0 < y p−1 , then we proceed by cases. If y p = 0, then y < q(p − 1)/p and py(p − 1) − pq(p − 2) < q. Thus (py p−1 )n+1 = 0. If y p > 0, then y > q(p − 1)/p which implies py p−1 = 1. Therefore, Eq(2p) holds for Łq+1 . COROLLARY 7.2.3. For n ≥ 3, and 1 < p < n, Łp+1 ∈ / W(n, p). For n ≥ 3 we define the equational class H(n) by the stipulation: \ H(n) = W(n) ∩ ( {W(n, p) | 1 < p < n and p is not a divisor of n}). THEOREM 7.2.4. For n ≥ 3, V(K n+1 ) = H(n). Proof. From Theorem 7.1.22 and Theorem 7.2.1, it follows that V(K n+1 ) ⊆ H(n). Now, we show the reverse inclusion. From Lemma 7.1.18, H(n) is a proper subvariety of MV. By Lemma 7.1.21, there exist two subsets P = {p1 , . . . , pr }, and N = {n1 , . . . , nt } of Z+ , with P ∪ N 6= ∅ and r, t ∈ Z+ , such that H(n) = V(Łp1 +1 , . . . , Łpr +1 , K p1 +1 , . . . , K pt +1 ). Since, by Lemma 7.1.17 and Theorem 7.1.22, K n+1 ∈ H(n), n ∈ {n1 , . . . , nt }; it is no loss of generality to assume n = n1 . From Lemma 7.1.18, we have nj ≤ n for every j ∈ {1, . . . , t}. Then, by Corollary 7.2.3, each nj divides n, whence H(n) = V(Łp1 +1 , . . . , Łpr +1 , K n+1 ). Again, by Corollary 7.2.3, H(n) = V(K n+1 ). COROLLARY 7.2.5. For n ≥ 3, V(K n+1 ) is characterized by the equations: ((n + 1)xn )2 = 2xn+1 , (pxp−1 )n+1 = (n + 1)xp for every positive integer 1 < p < n such that p is not a divisor of n.

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For n1 < · · · < nt , positive integers, assume that t ≥ 2, nt ≥ 5 and that ni is not a divisor on nj whenever i < j ≤ t. Let Σ denote the system of equations ((nt + 1)xnt )2 = 2xnt +1 , (pxp−1 )nt +1 = (nt + 1)xp , for every integer 1 < p < nt such that p is not a divisor of any ni , i = 1, 2, . . . , t, and let L(n1 , . . . , nt ) denote the variety defined by Σ. Moreover we define L(nt ) = H(nt ) if

t = 1,

L(2, 3) = W(3), L(3, 4) = W(4). THEOREM 7.2.6. For n1 < · · · < nt , positive integers, such that t ≥ 2, nt ≥ 5, V(K n1 +1 , . . . , K nt +1 ) = L(n1 , . . . , nt ). Proof. Firstly, we show that V(K n1 +1 , . . . , K nt +1 ) ⊆ L(n1 , . . . , nt ). We note that K ni +1 ∈ L(n1 , . . . , nt ) for every i ∈ {1, . . . , t}. If i = t, then the result follows from Theorems 7.1.17 and 7.1.22. If i < t, the result follows from Theorems 7.1.17 and 7.1.23, because p is not a divisor of any nt , i ∈ {1, . . . , t}. Now, by Lemma 7.1.21, we obtain L(n1 , . . . , nt ) = V(Łs1 +1 , . . . , Łsj +1 , K r1 +1 , . . . , K rg +1 ) for some positive integers s1 , . . . , sj , r1 , . . . , rg , and from Lemma 7.1.18, rg ≤ nt . Let q < nt be a positive integer not dividing any ni , i = 1, . . . , t. Choose y = q − 1 ∈ Lq+1 . Then qy q−1 = 1 and y q = 0. This implies that each element of the set s1 , . . . , sj , r1 , . . . , rg is a divisor of some index ni . Thus, we have L(n1 , . . . , nt ) ⊆ V(K n1 +1 , . . . , K nt +1 ). This completes the proof. COROLLARY 7.2.7. For n1 < · · · < nt , positive integers, assume that ni is not a divisor of nj , whenever i < j ≤ t. Then the subvariety V(Łn1 +1 , . . . , Łnt +1 ) is characterized by the identities: ((nt + 1)xnt )2 = 2xnt +1 , (pxp−1 )nt +1 = (nt + 1)xp , where 1 < p < nt and p does not divide any ni , i = 1, . . . , t. Let I = {α1 , . . . , αs } = 6 ∅, with α1 < · · · < αs , αi not dividing αj , whenever i < j ≤ s. Let J = {β1 , . . . , βt } with β1 < · · · < βt and I, J ⊂ Z+ . In case J 6= ∅, assume, for every i = 1, . . . , s, αi does not divide βj , for each j = 1, . . . , t. Furthermore, assume that αi is not a divisor of αj whenever i < j ≤ s, and βi is not a divisor of βj whenever i < j ≤ t. Let n = max{I ∪ J}. Let Σ0 denote the system of equations: ((n + 1)xn )2 = 2xn+1 , (pxp−1 )n+1 = (n + 1)xp ,

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for every positive integer p such that 1 < p < n and p does not divide i whenever i ∈ I ∪ J. Let Σ00 denote the system of equations: (n + 1)xq = (n + 2)xq , for every q such that q ∈

S

αi ∈I

∆(αi , J). Let H∗ be the variety defined by Σ0 ∪ Σ00 and

¯ = V(Łα +1 , . . . , Łα +1 , K β +1 , . . . , K β +1 ). H 1 n 1 m ¯ = H∗ . THEOREM 7.2.8. H Proof. The MV-algebras Łα1 +1 , . . . , Łαn +1 , K β1 +1 , . . . , K βm +1 are all members of H∗ . Thus by Corollary 7.2.3,Sthey verify all the equations of Σ0 . Furthermore, for each q q αi ∈ I, y ∈ Lαi +1 , and q ∈ αi ∈I ∆(αi , J), we have S either 0 = y or ord (y ) ≤ n. q q Thus, (n + 1)y = (n + 2)y for every q ∈ αi ∈I ∆(αi , J). This proves that Łα1 +1 , . . . , Łαn +1 verify Σ00 . Now let βh ∈ J and y ∈ KβhS+1 . Since q does not divide βh , then either y q = 0 or ord (y q ) ≤ n + 1, for every q ∈ αi ∈I ∆(αi , J). Thus, it follows that (n + 1)y q = (n + 2)y q . So, also K βj +1 satisfies Σ00 , for every βj ∈ J. ¯ ⊆ H∗ . Hence H By Theorem 7.1.22, H∗ is a proper subvariety, hence by Lemma 7.1.21, H∗ = V(Łq1 +1 , . . . , Łqh +1 , K p1 +1 , . . . , K pt +1 ) for q1 , . . . , qh , p1 , . . . , pt , h, t ∈ Z+ . Each index q1 , . . . , qh , p1 , . . . , pt , by Lemma 7.1.18, is smaller than or equal to n; while, by Corollary 7.2.3, it divides some element of I ∪ J. Hence, we infer that each MV-algebra Łq1 +1 , . . . , Łqh +1 , K p1 +1 , . . . , K pt +1 is a subalgebra of some MV¯ We algebra, Łα1 +1 , . . . , Łαn +1 , K β1 +1 , . . . , K βm +1 . Now we show that H∗ ⊆ H. ∗ will show that the greatest subalgebra of K αi +1 , which is member of H , is K αi +1 , for each ∈ I. Indeed, choose y = hαi − 1, zi ∈ Kαi +1 , for z ∈ Z+ ; we have y αi ∈ Rad (K αi +1 ) \ {0}; then (n + 1)y αi = (n + 1)(0, αi z) 6= (n + 2)y αi . Thus, the element y ∈ Kαi +1 is not a solution of the equation (n + 1)xαi = (n + 2)xαi of Σ00 . The thesis is now proved. We prove the main theorem of this section. THEOREM 7.2.9. Let W be a proper subvariety of MV. Then there exist finite sets I and J of integers ≥ 1 with I ∪ J 6= ∅ such that for any MV-algebra A the following are equivalent: (a)

A ∈ W,

(b)

A satisfies the equations: Eq(1) Eq(2) Eq(3)

((n + 1)xn )2 = 2xn+1 (px

p−1 n+1

)

with n = max{I ∪ J}, p

= (n + 1)x ,

q

(n + 1)x = (n + 2)xq ,

for every positive integer 1 < p < n such that p is not a divisor of any i ∈ I ∪ J S and for every q ∈ i∈I ∆(i, J).

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Proof. By Lemma 7.1.21 there exist integers α1 , . . . , αn , β1 , . . . , βm such that W = V(Łα1 , . . . , Łαn , K β1 , . . . , K βm ). By Theorem 7.2.8 Eq(1), Eq(2) and Eq(3) characterize W. THEOREM 7.2.10. Let A be an MV-algebra. Then the following are equivalent: (a)

A ∈ V(K n1 +1 , . . . , K nt +1 ),

(b)

A/Rad (A) ∈ V(Łn1 +1 , . . . , Łnt +1 ).

Proof. (a) ⇒ (b) ByQChang’s representation theorem, A can be subdirectly embedded into a direct product i∈I Ai where Ai = A/P for every P ∈ Spec(A). Let φ be the MV-embedding mapping defined s follows: φ(x) = (xi /Rad (Ai ))i∈I , where x = (xi )i∈I . By (j), for every i ∈ I, Ai /Rad (Ai ) ∈ V(K n1 +1 , . . . , K nt +1 ). Hence, by Proposition 7.1.5 rank(Ai /Rad (Ai )) = ki for some positive integer ki . But Ai /Rad (Ai ) is simple, and thus Ai /Rad (Ai ) ' Łki +1 . Therefore we obtain Łki +1 ∈ V(K n1 +1 , . . . , K nt +1 ). We can safely assume that n1 < · · · < nt , hence V(K n1 +1 , . . . , K nt +1 ) ⊆ V(K 2 , . . . , K nt +1 ) = W(nt ), by Lemma 7.1.18 ki = nt and Łki +1 ∈ V(Ł2 , . . . , Łnt +1 ) = T(nt ) by Lemmas 7.1.10 and 7.1.14. Assume ki not be a divisor of nt . Then, by Corollary 7.2.5 the following equation (ki xki −1 )nt +1 = (nt + 1)xki has to be true for Łki +1 . But this is not true, indeed. In fact, the element (ki − 1)/ki of Lki +1 does not satisfy the above equation. Hence ki is a divisor of nt . Then Ai /Rad (Ai ) can be embedded into Łnt +1 . Thus A/Rad (A) ∈ V(Łn1 +1 , . . . , Łnt +1 ). (b) ⇒ (a) Let P ∈ Spec(A) and Rad (A) ⊆ P , then by (jj), A/P ' (A/Rad (A))/(P/Rad (A)) ∈ V(K n1 +1 , . . . , K nt +1 ).

(∗)

If Rad (A) * P , then (A/P )/(Rad (A/P )) ' A/M , where M is the unique maximal ideal of A such that P ⊆ M . By (jj) and Proposition 7.1.5, A/M ' Łp+1 for some positive integer p and Łp+1 ∈ V(Łn1 +1 , . . . , Łnt +1 ). Assume again that n1 < · · · < nt , hence by Lemma 7.1.14, Łp+1 ∈ V(Łn1 +1 , . . . , Łnt +1 ) ⊆ V(Ł2 , . . . , Łnt +1 ) = T(nt ), by Lemma 7.1.10, p ≤ nt . Again, by Corollary 7.2.5, as above, we get p is a divisor of nt . Hence A/P can be embedded into Łnt +1 , hence (∗∗)

A/P ∈ V(Łn1 +1 , . . . , Łnt +1 ) ⊆ V(K n1 +1 , . . . , K nt +1 ).

By (∗) and (∗∗) we get A ∈ V(K n1 +1 , . . . , K nt +1 ).

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Historical remarks and further reading

MV-algebra theory has its origin in the study of infinite-valued Łukasiewicz propositional logic. The completeness theorem of this logic was firstly published by Rose and Rosser in 1958 [77]. An earlier proof given by Wajsberg was never published. MV-algebras were defined by Chang [12] in 1958, developing an algebraic version of Łukasiewicz propositional logic. He also gave an algebraic proof of the completeness theorem [13]. MV-algebras, therefore, stand in relation to the Łukasiewicz infinite valued logic as Boolean algebras stand in relation to classical 2-valued logic. Boolean algebras are not glued to their origin in logic. Their well-recognized use in other areas of mathematics led to an extensive investigation of their structure. The MV-algebra theory followed a similar development. Establishing connections with other areas of mathematics was a mainstream of investigation [4, 26, 59, 60]. In parallel, the comprehensive study of their intrinsic structure has shown the existence of a rich variety of classes of MV-algebras and this research continues in present. The study of MV-algebras was quiescent for about thirty years after their appearance. They have been considered with increasing attention after successful applications of Mundici in the study of AF C ∗ -algebras [59], lattice-ordered Abelian groups [60, 63] and of Belluce in Bold Fuzzy Set Theory [4]. In [60], Mundici proved the categorical equivalence between MV-algebras and lattice-ordered Abelian groups with strong unit. This theorem is the origin of a considerable number of remarkable results in the theory of MV-algebras. It turns out that the functor Γ plays a pivotal role in the theory of MV-algebras. In the same vein, a different categorical equivalence was established by Di Nola and Lettieri [26], showing that the category of lattice-ordered Abelian groups is equivalent to the category of perfect MV-algebras. The composition of the functor Γ with Grothendieck’s functor K0 yields a one-one correspondence between MV-algebras and a class of AF C ∗ -algebras. Such a correspondence was explored by Mundici in several papers, see, for example, [59] and [62]. In particular, the author introduced in [62], the AF C ∗ -algebra M1 corresponding to the free MV-algebra of one variable, rediscovered twenty years later by Boca in [9]. A first important class of MV-algebras was characterized by Chang [12]. He proved that the linearly ordered structures, MV-chains, are exactly quotients of MV-algebras with respect to the prime ideals. Successively, to cite some authors, Lacava [46], Belluce [4–6], Cignoli [17], Hoo [40] gave further contributions to the classification of MV-algebras. Further results concerning characterizations of classes of MV-algebras can be found in more recent papers, see for example [19, 23, 26]. MV-algebras form a variety, so the description of its subvarieties is an important issue. In [43] Komori gave a complete description of all subvarieties of MV-algebras. Concerning equational bases for such varieties, a first step was made by Grigolia [38, 39], who axiomatized equationally the class of Łn -algebras. With different approaches, Di Nola and Lettieri [27] and Panti [73] provided finite equational bases for each subvariety of MV-algebras.

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The representation theory of MV-algebras has a rich history. In [12], Chang proved the first representation theorem: any MV-algebra is isomorphic with a subdirect product of MV-chains. In the context of many-valued logic, the algebras of [0, 1]-valued functions have a special role. These are the semisimple MV-algebras, i.e. algebras without infinitesimals. Their functional representation was pointed out by Chang [12] and Belluce [4]. The representation of semisimple MV-algebras as algebras of continuous functions is proved in [18, 29]. A functional representation for all MV-algebras is due to Di Nola [21]: any MV-algebra can be seen as an MV-algebra of functions from a set to an ultrapower of [0, 1]. Another key result in the MV-algebra theory is McNaughton theorem [56], which led to a functional representation of free MV-algebras. Subsequent deep studies are due to Aguzzoli, Cignoli, Marra, Mundici and Panti. One can see [66] for a constructive proof of McNaughton Theorem and [72] for a geometrical proof. In the paper [67] the author opened the studies on states of MV-algebras. Quite recently, studies on states of MV-algebras received a renewed impulse, showing that such states play an analogous fruitful role for MV-algebras like probabilities do for Boolean algebras, connecting phenomena coming from groups, measure theory, topology and betting, being states a generalization of probabilities. In the last few years, the theory of states was studied by many experts in MV-algebras, see e.g. [44, 45, 74, 76]. K¨uhr and Mundici studied states using the notion of a coherent state by De Finetti with the motivation in Dutch book making, see [45]. The Kroupa-Panti theorem yields a one-one correspondence between states of an MV-algebra A and the regular Borel probability measures on the spectrum of maximal ideals of A, see [44, 74]. MV-algebraic probability theory is now a large area of studies which involves many topics, among them we find continuous states, conditional probabilities, Caratheodory algebraic probability, metric completion, axiomatics, entropy, complexity theory, invariant states, abstract Lebesgue integration, MV-algebraic ergodic theory, see [49, 53, 55, 68–70]. Of great interest are the recent studies on the category of finitely presented MValgebras. A combination of usage of the Γ functor and of methods coming from the theory of finitely presented Abelian `-groups and polyhedral geometry has been applied to reach extremely interesting results. Indeed, using the Γ functor, in [51] it is proved that finitely generated projective `-groups coincide with `-groups presented by a word only using the lattice operations. This strengthens a well known corollary of the celebrated Baker–Beynon duality: an Abelian `-group G is finitely presented (by a word only using the additive subtractive structure of G) if and only if it is finitely generated projective. On the other hand, projective unital `-groups and projective MV-algebras are rather the exception among finitely presented algebras (see [10, 11]). In his two papers [41, 42], Jeˇra´ bek relates projective MV-algebras with unification problems in Łukasiewicz logic. The non-commutative generalizations of MV-algebras were introduced, independently, by Georgescu and Iorgulescu [35], under the name pseudo MV-algebras, and by Rach˚unek [75], under the name generalized MV-algebras. The MV-algebraic operations ⊕ and are not necessarily commutative, so the new structures are categorically equivalent with lattice-ordered groups with strong unit, as proved by Dvureˇcenskij in [30]. Consequently, the corresponding propositional calculus developed in [48] has two implications and it provides a non-commutative approach to Łukasiewicz logic.

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Another research direction was to characterize the class of structures generated by [0, 1] in the language of MV-algebras enriched with the real product. These investigations led to the definition of PMV-algebras (product MV-algebras) [22, 57]. The analogue of Mundici’s theorem for these structures was obtained by Di Nola and Dvureˇcenskij [22]: there exists a categorical equivalence between PMV-algebras and lattice-ordered rings with strong unit. In [58], Montagna axiomatized the quasivariety generated by [0, 1] in the language of PMV-algebras. It seems quite natural to introduce “modules” over PMV-algebras. The structure of MV-module is defined and studied in [24, 47]. The MV-modules over [0, 1] are particularly interesting, since they are categorically equivalent with a class of Riesz spaces. A common generalization of MV-modules and product MV-algebras is obtained in [32], where MV-algebras with operators are defined. In [36] the author defines DMV-algebras and their corresponding propositional calculus, Rational Łukasiewicz logic. DMV-algebras are obtained by adding a countable family of operators to the structure of MV-algebra. Consequently, the Rational Łukasiewicz logic is an extension of the infinite valued Łukasiewicz calculus. MV-algebras are also intimately connected with other apparently remote domains of mathematics like toric varieties [1, 54], multisets [19], semirings [25, 37] or quantum structures [31]. One can also find in literature applications of Łukasiewicz logic to error correcting codes and fault-tolerant search [15, 16], based on the game-theoretic semantics for Łukasiewicz logic introduced by Mundici [64, 65] or connections with neural networks [2], image processing [28] and automata [37, 78]. Acknowledgements The authors acknowledge the referees, whose comments and suggestions improved the quality, the correctness and the form of our work. We especially mention their help in writing an up-to-date section on further readings. A grateful acknowledgment is also made to Petr Cintula and Carles Noguera for their help in obtaining a form that satisfies the standards of the present handbook. The second author has been partially supported by the Alexander von Humboldt Foundation through a return fellowship and partially supported by the strategic grant POSDRU/89/1.5/S/58852 cofinanced by ESF within SOP HRD 2007-2013.

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A NTONIO D I N OLA Department of Mathematics and Computer Science University of Salerno Via Ponte Don Melillo 84084 Fisciano (SA), Italy Email: [email protected] I OANA L EUS¸ TEAN Faculty of Mathematics and Computer Science University of Bucharest Str. Academiei 14, sector 1 C.P. 010014, Bucharest, Romania Email: [email protected]