SYMMETRIES ON ALMOST SYMMETRIC NUMERICAL SEMIGROUPS HIROKATSU NARI (NIHON UNIVERSITY)

Let N be the set of nonnegative integers. A numerical semigroup H is a subset of N which is closed under addition and N \ H is a finite set. Every numerical semigroup H admits a finite system of generators, that is, there exist a1 , ..., an ∈ H such that H = ha1 , ..., an i = {λ1 a1 + · · · + λn an | λ1 , ..., λn ∈ N}. We always assume that 0 ∈ H. We define F(H) = max{n | n 6∈ H} and g(H) = Card(N \ H). We call F(H) the Frobenius number of H, and we call g(H) the genus of H. We say that an integer x is a pseudo-Frobenius number of H if x 6∈ H and x + h ∈ H for all h ∈ H, h 6= 0. We will denote by PF(H) the set of pseudo-Frobenius numbers of H, and its cardinality is the type of H, denoted by t(H). If H = ha1 , a2 , . . . , an i, then we call a1 the multiplicity of H and denote it by m(H), and we call n the embedding dimension of H and denote it by e(H). For a numerical semigroup H with maximal ideal M = H \ {0}, we set M − M = {x ∈ N | x + M ⊆ M } and K = {F(H) − z | z 6∈ H}. We call M − M the dual of M and denote it by H ∗ , and we call K the canonical ideal of H. Definition 1. [BF] We say that a numerical semigroup H is almost symmetric if K \ H = PF(H) \ {F(H)}. Theorem 2. [Ba], [BF] Let H be a numerical semigroup with maximal ideal M . Then the following conditions are equivalent. (1) H is almost symmetric. (2) K ⊂ H ∗ . (3) 2 g(H) = F(H) + t(H). In this talk, we will characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. Moreover we give a criterion for H ∗ to be almost symmetric numerical semigroup. Our first result is the following theorem. Theorem 3. Let H be a numerical semigroup and let PF(H) = {f1 < f2 < · · · < ft(H) = F(H)}. Then the following conditions are equivalent. (1) H is almost symmetric. (2) z 6∈ H implies that either F(H)−z ∈ H or z = fi for all i ∈ {1, 2, ..., t(H)−1}. (3) fi + ft(H)−i = F(H) for all i ∈ {1, 2, . . . , t(H) − 1}. The following is the key lemma to prove our second result. 1

2

HIROKATSU NARI (NIHON UNIVERSITY)

Lemma 4. Let H be a numerical semigroup. Then F(H ∗ ) = F(H) − m(H). Theorem 5. Let H be an almost symmetric numerical semigroup. Then H ∗ is almost symmetric if and only if m(H) = t(H) + t(H ∗ ). Corollary 6. Let H be an almost symmetric numerical semigroup with t(H) ≤ 2. Then H ∗ is almost symmetric if and only if e(H) = m(H) − 1. References [Ba]

[BF] [RG]

V. Barucci, On propinquity of numerical semigroups and one-dimensional local Cohen Macaulay rings, Commutative algebra and its applications, 49-60, Walter de Gruyter, Berlin, 2009. V. Barucci, R. Fr¨ oberg, One-dimensional almost Gorenstein rings, J. Algebra, 188 (1997), 418-442. J. C. Rosales, P. A. Garc´ıa-S´anchez, Numerical semigroups, Springer Developments in Mathematics, Volume 20 (2009).

SYMMETRIES ON ALMOST SYMMETRIC NUMERICAL ...

Frobenius number of H, and we call g(H) the genus of H. We say that an integer x ... will denote by PF(H) the set of pseudo-Frobenius numbers of H, and its ...

139KB Sizes 1 Downloads 100 Views

Recommend Documents

On Distributing Symmetric Streaming Computations
using distributed computation has numerous challenges in- ... by these systems. We show that in principle, mud algo- ... algorithm can also be computed by a mud algorithm, with comparable space ... algorithms. Distributed systems such as.

On numerical semigroups
n-C 6 S' and n-(C+n~) r S' for all 0

On the Existence of Symmetric Mixed Strategy Equilibria
Mar 20, 2005 - In this note we show that symmetric games satisfying these ... mixed strategies over A, i. e. the set of all regular probability measures on A.

Linear Operators on the Real Symmetric Matrices ...
Dec 13, 2006 - Keywords: exponential operator, inertia, linear preserver, positive semi-definite ... Graduate Programme Applied Algorithmic Mathematics, Centre for ... moment structure and an application to the modelling of financial data.

on the minimal fourier degree of symmetric boolean ...
2. AMIR SHPILKA, AVISHAY TAL of course other areas of math and physics), a partial list includes learning theory, hardness of approximation, pseudo-randomness, social choice theory, coding theory, cryptography, additive combinatorics and more. A typi

Variations on the retraction algorithm for symmetric ...
With block methods get. 1) basic triangular shape. 2) super long columns. 3) short columns which don't fit into rank k correction or vanish. x x x x x x. x x x x x x x. x x x x x x x x. x x x x x x x x x r r r. x x x x x x x x x x r r.

Characterization of Partial Intrinsic Symmetries
We can further distinguish between global and partial symmetries. A global intrinsic symmetry maps .... Here we focus on research on detecting intrinsic symmetry. To compute infinitesimal ..... In: Symposium on Ge- ometry Processing (2012).

On the Almost Sure Limit Theorems IAIbragimov, MA ...
The statements about the convergence of these or similar distributions with probability one to a limit distribution are called almost sure limit theorems. We.

Choquet Integrals for Symmetric Ca
published online January 17, 2002 ... (i) the decision maker respects (Ak), (ii) f is a polynomial of degree k, (iii) the weight of all coalitions with ..... We then get: Tk.

Comparison of Symmetric Key Encryption Algorithms - IJRIT
In this paper we provides a comparison between most common symmetric key cryptography algorithms: DES, AES, RC2, ... Today it becomes very essential to protect data and database mostly in e-transaction. The information has .... For most applications,