Cancellative semigroup

About: Cancellative semigroup is a research topic. Over the lifetime, 1320 publications have been published within this topic receiving 13319 citations.

Isomorphism Theorems for Variants of Semigroups of Linear Transformations

Abstract: If S is a semigroup and a ∈ S, the semigroup (S, ◦) defined by x ◦y = xay for all x,y ∈ S is called a variant of S and (S, ◦) is denoted by (S,a). In 2003-2004, Tsyaputa characterized when two variants of the following transformation semigroups are isomorphic : the symmetric inverse semigroup, the full transformation semigroup and the partial transformation semigroup on a finite nonempty set. In this paper, we consider the semigroups under composition of all linear transformations of a finite-dimensional vector space over a finite field. We determine when its variants are isomorphic. We also obtain as a consequence in the same matter for the full n × n matrix semigroup over a finite field. Mathematics Subject Classification: 20M20, 20M10

On essential right congruences of a semigroup

Abstract: In ring theory one can give several approaches to the introduction of the concept of semisimplicity and a large number of equivalent formulations of this concept [5, 13]. Analogues of some of these formulations have been made, and studied, for semigroups [4, 6, 7, 8, 10, 11, 12] in terms of ideals or congruence relations. It seems possible that suitable and effective analogues can be made for each of these ring theoretical formulations in terms of congruence relations. However, unlike the situation for rings most of these analogues give inequivalent formulations in semigroups. One of the weakest of these is the nonexistence of a proper, essential right congruence. A right congruence ~ in a semigroup S is essential if for any right congruence a we have Q N a = z (the identity relation) implies a = z. The main result of this paper is a characterization of semigroups with the d.c.c, on right ideals and having no proper essential right ideals and having no proper essential right congruences. The first step in this characterization is a description of the lattice of right ideals in such a semigroup. Our results of this description in Section 2 should be compared with the work of Feller and Gantos [2] and Fountain [3]. While the class of semigroups studied in these two papers are defined quite differently than the class in this paper there is a striking similarity in the results; thus, suggesting a common area of investigation. In the subsequent sections the main technique used is the examination of a selection of proper right congruences and the implications on the multiplicative properties obtained from the assumption of nonessentiality.

Representations of certain compact semigroups by -semigroups

Abstract: An HL-semigroup is defined to be a topological semigroup with the property that the Schutzenberger group of each XI-class is a Lie group. The following problem is considered: Does a compact semigroup admit enough homomorphisms into HL-semigroups to separate points of S; or equivalently, is S isomorphic to a strict projective limit of HL-semigroups? An affirmative answer is given in the case that S is an irreducible semigroup. If S is irreducible and separable, it is shown that S admits enough homomorphisms into finite dimensional HL-semigroups to separate points of S. Introduction. A semigroup analog of the theorem of Peter and Weyl is not in existence at the present time. Indeed, such a theory for compact semigroups closely paralleling that for compact groups is generally believed to be unfeasible. In this paper an alternate approach is considered. The alternative is to replace groups of nonsingular complex matrices by compact semigroups with the property that the Schiutzenberger group of each X-class is a Lie group. Such semigroups are called HL-semigroups. The following question is considered: Given a compact semigroup S, do there exist enough homomorphisms of S into HL-semigroups to separate the points of S? Furthermore, can these homomorphisms be chosen so as to preserve the XV'-class structure of S? We follow the current trend and call a homomorphism from S into an HL-semigroup T a representation of S by the HLsemigroup T. The main result of this paper is that every irreducible semigroup admits enough X#'-class separating representations by HL-semigroups to separate points in S. Moreover, if S/I is separable, then each of the HL-semigroups may be chosen to have finite dimension. ?1 is devoted to preliminary results of a general nature. ?2 deals with irreducible semigroups. For the most part we will use the terminology and notation of [7]. All semigroups, homomorphisms, and isomorphisms will be in the category of compact semigroups and continuous homomorphisms. The authors are indebted to J. D. Lawson, Michael W. Mislove, and Eleanor Bailey for their useful comments and suggestions. Received by the editors January 30, 1969 and, in revised form, August 11, 1969. AMS Subject Classifications. Primary 2205; Secondary 2092, 2210, 2250.