THE THEORY OF SEMIGROUPS L. M. GLUSKIN Istoriya Otetschestvennoy Matematiki, Kiev, 3 (1968), 321–332

The study of operations, a fundamental notion in algebra, has in turn led to the study of various types of ‘generalised groups’: groupoids, multigroups, etc. On the whole, these objects have played only a sporadic role in algebra. Of them, only two — semigroups (algebraic systems with a single associative binary operation1) and, to a lesser extent, quasigroups (algebraic systems with a single invertible binary operation) — have become objects of deep and widespread investigation. The present theory of semigroups has hundreds of papers devoted to it in Soviet mathematical literature, and thousands in the world. The living theory of semigroups has deep connections with a whole series mathematical disciplines, not only comparatively long-standing ones (geometry, ring theory, functional analysis, number theory, and others), but also with more recent theories (graph theory, the theory of algorithms, the mathematical theory of automata). The theory of quasigroups has turned out to have natural connections with geometry and combinatorial analysis. The first deep results in semigroup theory were obtained in the 1920s by the distinguished Soviet algebraist and professor of Kharkov University, A. K. Suschkewitsch; he also undertook the first investigations in the theory of quasigroups. A significant number of Suschkewitsch’s results were collected in his monograph The theory of generalised groups (1937).2 Until his work, there seems to have been only one published article on semigroups — a short paper by Dickson on cancellative semigroups (1905). Of Suschkewitsch’s results, we mention first the investigation of minimal twosided ideals in semigroups; these are often called ‘kernels’, or ‘Suschkewitsch kernels’, in the literature. He showed that every finite semigroup possesses a kernel K, formed from the union of a family of pairwise-disjoint isomorphic groups Gij . Later, in the 1940s, Rees, Clifford, Schwarz, and others, developed an important division theory for semigroups — the theory of completely simple semigroups, Date: January 21, 2009. Please note that blue italic text indicates an uncertain translation. 1 In the first (Russian) works in which they appeared, semigroups were termed ‘generalised associative groups’, ‘associative systems’ or sometimes ‘groupoids’; by ‘semigroups’ were meant algebraic systems which are now termed cancellative semigroups (see below). 2 A detailed bibliography for semigroups is given in E. S. Lyapin’s monograph Semigroups, and also in the surveys on semigroups [1–3]. 1

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i.e., semigroups with no proper two-sided ideals, but containing minimal left and right ideals. Suschkewitsch’s work marked the beginning of a highly fruitful trend in semigroup theory — the investigation of semigroups which are in some way ‘close’ to groups. In particular, he studied the structure of arbitrary finite semigroups which are formed from the union of two disjoint groups. Subsequently, Clifford, Croisot, Tamura, L. N. Shevrin, and others, studied the structure of semigroups formed from the disjoint union of groups or of other semigroups. As a measure of the ‘closeness’ of an arbitrary semigroup S to a group, we can solve the following equations ax = b,

(1)

ya = b,

(2)

for different a, b ∈ S, and determine how many distinct solutions each equation has. Suschkewitsch studied cancellative semigroups, i.e., semigroups in which each of equations (1) and (2) has no more than one solution. He also investigated distinct classes of semigroups with one-sided cancellation, i.e., semigroups with a unique solution to only one of the equations (1) or (2). Out of all his results, we make special mention of those concerning semigroups S in which equation (1) has no more than one solution for each a, b ∈ S, but equation (2) has at least one solution, that is, semigroups with left division and left cancellation. Suschkewitsch considered direct products of semigroups (1941). He also devoted a series of papers to semigroups of transformations and matrix semigroups (in particular, representations of semigroups by transformations and by matrices). A significant contribution to the development of semigroup theory was made by A. I. Malcev in his results (1937–1940) on the embedding of semigroups in groups. He found necessary and sufficient for a cancellative semigroup to be embedded isomorphically in a group; he proved that there is no finite system of elementary axioms for a semigroup to be embedded in a group; he described all the non-isomorphic minimal groups in which a given cancellative semigroup may be embedded. The study of semigroups which may be embedded in groups was subsequently continued by other researchers. Apart from the results of A. K. Suschkewitsch and A. I. Malcev in the USSR, and also a small number abroad, very few results on semigroups appeared until the 1940s, and then only isolated results. Not yet widespread, what little work there was had to be abandoned during the Second World War and was resumed at the start of the 1950s. In the Soviet Union, this was chiefly by E. S. Lyapin and his, then still very few, students. Lyapin moved away from his original group-theoretic subject-matter and, together with his students, set about the investigation of semigroups. Up to 1950, he considered questions connected with homomorphisms of semigroups, introduced the important concepts

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of normal complexes and normal subsemigroups, and studied simple (i.e., possessing no nontrivial homomorphisms) and semisimple commutative semigroups. In 1953, he introduced the notion of a magnifying element in a semigroup and studied invertibility of elements in semigroups, i.e., solvability of equations (1) and (2). He explained the connection between invertibility of elements in semigroups and the presence of magnifying elements, and considered the structure of monoids containing magnifying elements. Later, these results were generalised to semigroups possessing only a one-sided identity. In 1954, Lyapin introduced the concept of a basis class for a class of semigroups Γ, i.e., a minimal subclass Γ0 ⊆ Γ such that every semigroup S ∈ Γ is a union of subsemigroups from Γ0 . In 1953, he began to consider semigroups of transformations. E. S. Lyapin runs a seminar on semigroups at the Leningrad Pedagogical Institute. Its participants began in the 1950s by studying ideals in semigroups (in particular, semigroups with the minimal condition for principal left ideals), structural properties of semigroups, free and direct products of semigroups (and other constructions), and finite holoidal semigroups (i.e., semigroups ordered by the relation of divisibility). They later turned to the systematic study of semigroups of transformations. The seminar was for a long time the only such in the country to deal with semigroup-theoretic problems, but it didn’t lose its significance when similar seminars were later introduced at other centres. At the start of the 1950s in Saratov, the famous Soviet geometer V. V. Wagner began his fundamental consideration of inverse semigroups and representations of semigroups. In 1952–3, A. I. Malcev returned to semigroup-theoretic subject-matter and studied, in particular, semigroups of transformations and matrix semigroups. In Sverdlovsk, P. G. Kontorovich, and some of his students, investigate problems connected with semigroups. L. M. Gluskin, a student of A. K. Suschkewitsch, works in Kharkov. He has studied simple semigroups with zero which contain minimal one-sided ideals (1955), has found homomorphisms from semigroups with left inversion into groups (1955), and has found a further simple description of homomorphisms of completely simple semigroups (1956). These results were later reproduced, independently of Gluskin, by Tamura, Saito and Munn. Gluskin has considered (1956) the possibility of extending homomorphisms of semigroups (in particular, periodic semigroups), has found a basis class for inverse semigroups (1957), and has studied matrix semigroups and semigroups of transformations (since 1954). In Tartu in 1956, Ya. V. Hion (a student of A. G. Kurosh) considered ordered semigroups. It is clear that in the 1950s, the theory of semigroups had grown into an independent area of algebra with its own problems and methods, and monographs devoted to the theory began to appear. Of these, the first in the world was E. S. Lyapin’s book “Semigroups”, which played a significant role in the development of the theory. Up until this time, there were noticeable differences in the approach, methods, constructions and terminology of the published works on semigroups. Lyapin’s book was the first to set down, in a coherent form, the

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basic trends of the algebraic theory of semigroups, putting forward the general point of view and outlining some prospects for development. At this time, E. S. Lyapin established (and still leads) the Leningrad school of semigroup theory. His students work in different centres around the country: A. Ya. A˘ızenshtat and J. C. Ponizovski˘ı in Leningrad, K. A. Zaretski˘ı in Novosibirsk, L. B. Shneperman in Minsk, E. G. Shutov in Taranrog, and others. We now discuss the Saratov school of semigroup theory. The founder and leader of the school is V. V. Wagner. Of his students, we pick out first and foremost B. M. Schein and A. E. Liber. This school is notable for its wide application of set-theoretic and logical methods, mainly via the theory of binary relations, in which Wagner has made an essential contribution. Both schools are very similar in their approach (about which we will speak later) and have close ties. The theory of semigroups arose when other areas of algebra — the theory of rings, the theory of groups and, to a lesser extent, the theory of lattices — had reached a certain completeness and harmony. Naturally, when semigroup theory first appeared, the semigroup questions considered by many authors were analogous to well-known solved problems in the theories of groups and rings. The influence of ring theory, for example, can be seen in the study of regular semigroups and, in particular, of completely simple semigroups, mentioned above. Observe also the reverse process: there has appeared a paper on associative rings, in which the strong influence of the theory of semigroups is remarked upon. Thus V. I. Schneidm¨ uller (Magnetogorsk, 1961) has studied the question of the existence of magnifying elements in multiplicative groups of rings. L. M. Gluskin (1959) obtained a characterisation of simple rings with minimal one-sided ideals in terms of completely simple ideals of their multiplicative semigroups. He (1960) and L. A. Skornjakov (later) studied rings in which the ideals of the multiplicative semigroup form ideals of the ring; this work is connected with results of Aubert and Brameret. S. R. Kogalovski˘ı (1961) proved that the class of multiplicative semigroups of rings and multiplicative semigroups of fields do not form axiomatised classes. A. I. Malcev (1953) considered semigroups which are analogues of nilpotent groups. P. G. Kontorovich (1953–1956) studied properties of subsemigroups of groups. Since 1956, J. S. Ponizovski˘ı has studied matrix representations of arbitrary semigroups, and, in connection with these, semigroup rings (his results are connected with the investigation of matrix representations by Clifford, Munn and others). A. Kh. Livshits (Moscow, 1960) built a general theory of direct decompositions of idempotents in semigroups, uniting the theory of direct decompositions in categories and the corresponding lattice theory, and, lastly, going back to classical questions about isomorphic direct decompositions in groups and rings. One of the representative trends of semigroup theory is the development of group-theoretic ideas by P. G. Kontorovich’s pupil L. N. Shevrin (Sverdlovsk). His work began appearing in print in 1960. He investigated semigroups with different finiteness conditions (semigroups with a minimal condition for subsemigroups,

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periodic semigroups with a maximal condition for subsemigroups, etc.), often reducing their description to group-theoretic questions. He has already proved a theorem about local finiteness connected with locally finite semigroups, a theorem about locally nilpotent semigroups with the idealiser condition (an analogue of a group-theoretic result of B. I. Plotkin), and found necessary and sufficient conditions for a subsemigroup of a free semigroup to be free (subsemigroups of free semigroups have also been investigated by Sch¨ utzenberger, Cohn and others). L. N. Shevrin and J. S. Ponizovski˘ı devoted some work to the study of the structure of semigroups by means of ideal chains. Using this, Shevrin has found, in particular, criteria for a semigroup to be nilpotent. Results from the lattice theory of groups have undoubtedly given rise to the investigation of lattice properties of semigroups. We recall the appropriate definitions. Let Σ(S) be the lattice of all subsemigroups of S. An isomorphism from the lattice Σ(S) onto another such lattice Σ(S 0 ) is called a lattice isomorphism of the semigroup S to the semigroup S 0 . In particular, if there exists a (semigroup) isomorphism or anti-isomorphism ϕ from a semigroup S to a semigroup S 0 , then it naturally gives rise to a lattice isomorphism from S to S 0 . We say that S is structurally defined in a class of semigroups K if every semigroup S 0 ∈ K is lattice isomorphic to S (relative to the operations in these semigroups). The development of a lattice theory for semigroups was initiated by the work of R. V. Petropavlovskaya, a student of E. S. Lyapin (Leningrad). She proved in particular that every free semigroup and every nonperiodic Abelian group is structurally defined in the class of semigroups, and that every semigroup that is lattice isomorphic to a nonperiodic group is a group (1951). She also gave a complete description of semigroups which are structurally isomorphic to some group (1956–1957). Some classes of groups which are structurally defined (in the above sense) in the class of all groups were isolated at the end of the 1950s and the start of the 1960s by students of P. G. Kontorovich. Since 1961, deep advances in the study of lattice properties of semigroups have been made by L. N. Shevrin. For semigroups of some class, he has classified the lattice isomorphisms of these semigroups and found structural definitions in this class of semigroups. This was for commutative nonperiodic semigroups with cancellation, commutative semigroups of idempotents, rectangular semigroups, orderable semigroups with cancellation, etc. Shevrin gave a complete description of semigroups S with Σ(S) a Dedekind or distributive lattice (the latter results ´ were found simultaneously and independently by M. Ego), semigroups for which Σ(S) forms a lattice with relative complements, a lattice with unique complements, and semigroups with the lattice Σ(S) of finite breadth, etc. In terms of the lattice Σ(S), he characterised some classes of semigroups of idempotents and groups: groups without torsion, nonperiodic, Abelian nonperiodic, ordered, etc. He initiated the investigation of basic lattice properties of semigroups and,

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in particular, found a number of lattice-axiomatised classes of semigroups and proved the unsolvability of the elementary theory of the lattice of subsemigroups. In work of A. A. Markov in 1947 and 1951 (and also in work of E. Post, carried out independently of Markov), semigroups were found to be one of the first objects of study in the theory of algorithms. Finitely presented semigroups (and, in particular, algorithmic problems in them) were later studied by S. I. Adyan (1955 and 1960) and others. A student of A. I. Malcev, M. A. Taitslin (Novosibirsk, 1962), proved the unsolvability of the elementary theory of commutative, cancellative semigroups. Identities in semigroups were studied by S. I. Adyan (1962), A. P. Biryukov and others; defining relations by A. Ya. A˘ızenshtat and others. Restrictive semigroups (i.e., idempotent semigroups satisfying the identity xyz = yxz) were studied by V. V. Wagner (1962) and B. M. Schein (1963) in connection with considerations of partial mappings of sets. B. M. Schein proved that the class of inverse semigroups (see below), considered as involutive semigroups, are primitive but may not be defined by a single identity. The most natural source of problems in semigroup theory is the study of semigroups of transformations. It is well-known that the set S(Ω) of all full (everywhere-defined) transformations (self-mappings) of an arbitrary set Ω is a semigroup with respect to superposition of transformations. A. K. Suschkewitsch showed that every finite abstract semigroup may be represented isomorphically as a semigroup of single-valued transformations of the corresponding set. This result was subsequently extended to arbitrary semigroups; besides the semigroup S(Ω), semigroups of partial and multiplace transformations have also been used to represent semigroups. It is natural that the objects of study of Soviet algebraists (A. K. Suschkewitsch, A. I. Malcev, E. S. Lyapin, V. V. Wagner and their students, L. M. Gluskin and others) are semigroups of transformations, and are, first and foremost, by strength of their universal character, “symmetric” semigroups S(Ω) of all full transformations of a set Ω. Other “symmetric semigroups” have also been studied: the semigroup W (Ω) of all partial transformations of a set Ω; V (Ω) of all its one-one partial transformations; P (Ω) of all its multiplace transformations (binary relations); H(Ω) of all its full one-one transformations, and others. Ideals, ideal layers, automorphisms, normal subsemigroups, defining relations, inner characterisations, etc. have been studied for each of these semigroups. The themes of the Leningrad and Saratov schools of semigroup theory (and also those of L. M. Gluskin) are mainly connected with semigroups of transformations. They consider the theory of semigroups to be the abstract study of the superposition of transformations of the same general form — not necessarily invertible or everywhere-defined, or even single-valued (just as the theory of groups is the abstract theory of invertible transformations). The paramount role that such an approach has acquired reflects the fact that the study of “abstract” semigroup properties is important for other areas of mathematics, since semigroups arise

THE THEORY OF SEMIGROUPS

7

naturally in algebra and other areas through their representations as semigroups of transformations. One of the most successful examples of such a set-theoretic approach in the theory of semigroups is that of inverse semigroups. The concept of an inverse semigroup was introduced by V. V. Wagner in 1952 (he called them generalised groups) in connection with the foundations of differential geometry. This is a regular semigroup (i.e., a semigroup in which for each element x there exists an element x0 such that xx0 x = x and x0 xx0 = x0 ) with pairwise commuting idempotents (somewhat later, but apparently independently of V. V. Wagner, the notion of an inverse semigroup was introduced by Preston, and afterwards also arrived at by Ehresmann). The particular importance of inverse semigroups, first and foremost, is that they form an abstract class of algebras which are isomorphic to semigroups of one-one partial transformations. Inverse semigroups provide a class of objects which are interesting in their own right and are objects of investigation in the USSR and abroad. Fundamental results in the theory of inverse semigroups were obtained by V. V. Wagner and his students, chiefly A. E. Liber and B. M. Schein. B. M. Schein connected the theory of inverse semigroups with that of the inductive groupoids of Croisot–Ehresmann. E. S. Lyapin (1953) introduced the concept of a densely embedded ideal of a semigroup. Densely embedded ideals were used by him and other investigators to obtain an abstract characterisation of semigroups of transformations. Properties of densely embedded ideals in abstract semigroups were studied by L. M. Gluskin (1960–1963), L. N. Shevrin (1960) and E. S. Lyapin (1962). In particular, they investigated the conditions for there to exist a semigroup S which contains a given semigroup A as a densely embedded ideal, thereby connecting dense embedding with semigroups of translations, by means of which an isomorphism of the densely embedded subsemigroup A ⊆ S may be extended to an isomorphism of the semigroup S. The concept of dense embedding may be generalised to arbitrary structure in the sense of Bourbaki. V. V. Wagner (1956) described the representation of an arbitrary abstract semigroup S by (partial) transformations of a some set (he described exactly the construction by means of which every such representation may be realised). E. S. Lyapin (1960) found other constructions for realising this representation in which the embedding of a semigroup S in an oversemigroup is used. B. M. Schein described (1962) all representations of a semigroup S by binary relations (multiplace transformations). He has already obtained deep and general results about semigroups of multiplace transformations, so that we can already speak of a mature theory of semigroups of binary relations (the only such results before Schein’s were some isolated results obtained by K. A. Zaretski˘ı in 1958–1959). In particular, from the results of Schein on semigroups of multiplace transformations, using only “inner” properties of the semigroup S, we can obtain a new description of all representations of a semigroup S by transformations (1961). Schein currently studies transitive representations of semigroups of transformations (these were

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studied, independently of him, by Tallini), so-called symmetric representations, and representations by which the “regular” quasi-order relation between elements of a semigroup S, connected with the semigroup operation, turns into a highly natural relation between transformations. The existence for an inverse semigroup of a representation by means of oneone partial transformations was first proved by V. V. Wagner (1953). These representations were also studied by Preston (1954). B. M. Schein (1961–1962) described the structure of all representations of inverse semigroups by one-one partial transformations. The methods developed by Schein allowed him also to solve other problems concerning inverse semigroups. In particular, he found a new class of representations of inverse semigroups by binary relations of a special form. Problems about the isomorphism of representations of semigroups of transformations of this or other form have great significance in the theory of semigroups when it comes to questions about embedding of semigroups. Above, we spoke about densely embedded semigroups and about the solution by A. I. Malcev of problems of embedding semigroups in groups. In 1960, B. M. Schein solved related problems: he found necessary and sufficient conditions for semigroups to be embedded in inverse semigroups. This result is equivalent to the description of the class of semigroups which admit isomorphic representation by one-one partial transformations. Semigroup embeddings were studied by E. S. Lyapin and his students, mainly ´ G. Shutov. Shutov investigated questions about the possibility of embedding a E. semigroup S in a semigroup without proper ideals and in a complete semigroup. He also considered questions about the possibility of embedding a semigroup S in a semigroup S 0 possessing given properties for some elements of the semigroup S, for example invertibility — solvability of equations (1) and (2). There exist very close links between the theory of semigroups and the mathematical theory of automata. In particular, every automaton A may be interpreted as a semigroup of transformations of the set A of its states — indeed, as the representation of the free semigroup over its input alphabet in the semigroup of A. Therefore, many theorems in the theory of semigroups may be transformed into rich theorems in the theory of automata. On the other hand, there are problems in the theory of automata which have been solved with the help of the apparatus of the theory of semigroups. It is of interest for the algebraic theory of semigroups that the solutions of (semigroup) problems are emerging naturally from the abstract theory of automata. This work is only beginning. The exploration of the ties between automata and semigroups occupies V. M. Glushkov (Kiev) and some of his students. In his book “Semigroups”, E. S. Lyapin considered semigroups of endomorphisms of a set together with an arbitrary collection of relations (speaking generally, infinitely many). Later he called such a set a “system of general form”.

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By endomorphisms of such a system A we understand full (later, partial) transformations of a set A which do not destroy the relations of the system A. Lyapin emphasised the role of semigroups of endomorphisms: they permit the description of ties between the original relations of the system A and algebraic properties of its semigroups of endomorphisms. “It is natural to expect”, he wrote, “that the study of semigroups of endomorphisms and the establishment of the abovementioned connections between their properties and the properties of the sets themselves should be one of the most important directions of development in semigroup theory and should very likely be particularly rich in applications of the algebraic theory of semigroups in various branches of mathematics” (p. 41). The book of E. S. Lyapin was still at the printers when there appeared a series of articles (1959–1960) by L. M. Gluskin, working at this time in Communarsk (Donbass), about semigroups of endomorphisms of mathematical structures in the sense of Bourbaki. These works are the beginning of the realisation of the description of the above programme. In them, besides investigating the formal algebraic character (in particular, with the notion of endomorphism interpreted so that it is contained in that of continuous transformation of a topological space), Gluskin revealed deep ties between the theory of semigroups and other mathematical theories: graph theory, topology, linear algebra and others. It turned out, for example, that semigroups of all endomorphisms of a quasi-ordered set define this set exactly up to isomorphism or duality. (Subsequently, analogous results about semigroups of full and partial endomorphisms of graphs were obtained, chiefly by students of E. S. Lyapin and L. M. Gluskin.) Likewise, semigroups of endomorphisms of a linear space of rank not less than 2 define this space exactly up to isomorphism; some semigroups of homeomorphic transformations of a topological space A, considered as an algebraic semigroup (without the induced topology), in a series of cases define this space exactly up to homeomorphism. Significant results about semigroups of continuous transformations of a topological space were obtained subsequently by L. B. Shneperman. He showed, in particular, that the (algebraic) semigroup C(A) of all continuous transformations of a full regular topological space A, containing a simple curve, defines this space precisely up to homeomorphism. Some properties of graphs and of linear and topological spaces in the work of Gluskin and other investigators of algebraic characterisations are in terms of the corresponding semigroups. V. V. Wagner and B. M. Schein obtained abstract characterisations of some relations (in particular, the relation of set-theoretic inclusion) in semigroups of partial transformations. These (and other) aspects of investigation led Wagner (1960), Gluskin (1960), Plotkin (1963) and Schein (1965–1966) to the study of semigroups given in terms of the structures of Bourbaki. Schein introduced the (in semigroup theory, important) notion of the algebra of relations. Wagner and Schein took interest in investigation by means of the theory of involutive semigroups. Lyapin studied the compatibility of ordering with superposition in

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semigroups of transformations and the maximum compatible ordering in abstract semigroups. Topological semigroups are studied by students of E. S. Lyapin, L. B. Shneperman and P. D. Kruming. To Shneperman, in particular, are due some interesting results on the representation of topological semigroups of continuous transformations of a topological space. Since 1952, V. V. Wagner (and later also his students) has studied semiheaps — algebraic systems S with a single ternary operation [s1 s2 s3 ] = s satisfying an associativity-type identity [[s1 s2 s3 ]s4 s5 ] = [s1 [s4 s3 s2 ]s5 ] = [s1 s2 [s3 s4 s5 ]] . For applications in the foundations of differential geometry, important semiheaps turn out to be those with a special condition of commutativity — so-called generalised heaps. Generalised heaps are a class of algebraic systems which give an “abstract” description of superposition of one-one partial mappings of one set to another (just as semigroups describe superposition of mappings of a set into itself). Thus the theory of generalised heaps has close ties with the theory of inverse semigroups. Wagner, Gluskin and Schein revealed very far-reaching analogues between the general theory of semigroups and the theory of semiheaps. The sources of these analogues were elucidated at considerable length by Gluskin by means of the study of so-called positional operatives — an algebraic system with a single n-ary operation, satisfying a special associativity-type identity. V. V. Wagner studies other such algebraic systems close to semigroups and having applications to geometric questions. Characters of semigroups (their homomorphisms into the multiplicative semigroup of complex numbers) were studied by M. M. Lesohin (1960–1962). These ˇ Schwarz, R. Hunter and others. Lesohin investigations are close to a trend of S. considered generalised characters of commutative semigroups — their homomorphisms into other commutative semigroups — and questions of approximations, i.e., subdirect products of semigroups. Such subdirect products are studied by B. M. Schein. By means of a natural extension of the theory of characters, Lesohin developed the theory of systems with exterior multiplication — triples {A, B, C} of commutative semigroups, in which every a ∈ A and b ∈ B define an element a ◦ b ∈ C, moreover a ◦ b and A ◦ b are homomorphisms of a and, correspondingly, of A into C. The theory of semigroups continues to develop successfully in the USSR. The investigation is deepening, the number of works increasing. It is therefore natural that in drawing this survey, we have not considered more recent works. This survey relates to the algebraic theory semigroups; we have not mentioned the investigation of numerical semigroups and semigroups of operators in Banach spaces, and have only very briefly described the work on ordered semigroups.

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References 1. L. M. Gluskin, Semigroups (Russian), Itogi Nauki. (Algebra. Topol. 1962) 1 (1963), 33–58. , Semigroups (Russian), Itogi Nauki. Ser. Mat. (Algebra. 1964) 3 (1966), 161–202. 2. 3. L. M. Gluskin, B. M. Schein, and L. N. Shevrin, Semigroups (Russian), Itogi Nauki. Ser. Mat. (Algebra. Topol. Geom. 1966) 5 (1968), 9–56.

Translated by CHRISTOPHER HOLLINGS

THE THEORY OF SEMIGROUPS The study of operations, a ...

The study of operations, a fundamental notion in algebra, has in turn led to the study of various types of 'generalised groups': groupoids, multigroups, etc. On.

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Cochin University of Science and. Technology ... Abstract. We report a photoacoustic (PA) study of the thermal and ... The ex- perimental data obtained from the measurement of the PA signal as a ..... working towards his PhD degree at the In-.

A computational study of the mechanical behavior of ...
E-mail address: [email protected] (L. Anand). .... with G and K the elastic shear modulus and bulk modulus, ..... Application of the model to nanocrystalline nickel.

A study of the formation of microporous material SAPO-37
Mar 29, 2013 - diffraction (PXRD). Scanning electron microscopy (SEM) was uti- lized to observe the morphological changes. Further, the nucleation and crystal growth were examined by atomic force microscopy. (AFM). The combination of these techniques

a computational study of the characteristics of ... - Semantic Scholar
resulting onset of non-survivable conditions which may develop within the aircraft passenger cabin. To satisfy ... related applications. In a recent application of fire field modelling [11, 12], Jia et al used the CFD fire simulation ..... predicted

The Modernization of Education: A Case Study of ...
a solid background in both European languages and the sciences.5 It held the same ..... main focus of this school was to educate young men from top families to be .... Tunisia, from the data released by the Tunisian government, had made a bit.

THE TAMING OF THE ID A STUDY OF.pdf
There was a problem previewing this document. Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. THE TAMING ...

the crowd: a study of the popular mind.pdf
the crowd: a study of the popular mind.pdf. the crowd: a study of the popular mind.pdf. Open. Extract. Open with. Sign In. Main menu. Displaying the crowd: a ...

The Theory of Higher Randomness
Jul 12, 2006 - Sacks's result that every nontrivial cone of hyperdegrees is null. Measure-theoretic uniformity in recursion theory and set theory. Martin-Löf's ...

On Apéry Sets of Symmetric Numerical Semigroups
In this paper, we give some results on Apéry sets of Symmetric. Numerical Semigroups with e(S)=2. Also, we rewrite the definitions n(S) and H(S) by means of ...