Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

BYD. B. MCALISTERAbstract. A regular semigroup S is locally inverse if each local submonoid eSe, ean idempotent, is an inverse semigroup. It is shown that every locally inversesemigroup is an image of a regular Rees matrix semigroup, over an inversesemigroup, by a homomorphism 0 which is one-to-one on each local submonoid;such a homomorphism is called a local isomorphism. Regular semigroups which arelocally isomorphic images of regular Rees matrix semigroups over semilattices arealso characterized.

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Rees matrix covers for locally inverse semigroups

A regular semigroup S is called a pseudo-inverse semigroup if eSe is an inverse semigroup for each e= e2 C S. We show that every pseudo-inverse semigroup divides a semidirect product of a completely simple semigroup and a semilattice. We thereby give a structure theorem for pseudo-inverse semigroups in terms of groups, semilattices and morphisms. The structure theorem which is presented here generalizes several structure theorems which have been given for particular classes of pseudo-inverse semigroups by several authors, and thus contributes to a unification of the theory. Completely (0-) simple semigroups and inverse semigroups form the first prototypes for the study of pseudo-inverse semigroups. We therefore can say that the theory of regular semigroups began with the study of pseudo-inverse semigroups [40, 45]. We may distinguish four successful trends in the papers which since then have dealt with some wider classes of pseudo-inverse semigroups: 1. the subdirect products of completely 0-simple and completely simple semigroups, 2. the generalized inverse semigroups (orthodox pseudo-inverse semigroups, 3. the normal band compositions of inverse semigroups, and 4. Rees matrix semigroups over inverse semigroups (with zero). Subdirect products of completely 0-simple semigroups and completely simple semigroups were initiated in [13, Chapter 2] and studied in great detail in [18] (see also ?4 of [14]); this class contains several interesting subclasses: (a) the trees of completely 0-simple semigroups [18] which include the primitive regular semigroups [7, Vol. II, 16, 39, 44, 46], (b) the regular locally testable semigroups [50] which include the normal bands [36] and the combinatorial completely 0-simple semigroups, (c) the normal bands of groups [37] which include the semilattices of groups [7, Vol. I], (d) the subdirect products of Brandt semigroups which include the locally testable semigroups which are inverse semigroups [50] and the primitive inverse semigroups [39]. The generalized inverse semigroups were introduced in [48] as a special class of orthodox semigroups; they include (a) the inverse semigroups, (b) the orthodox completely 0-simple semigroups [9] and the rectangular groups, (c) the Received by the editors October 25, 1979 and, in revised form, March 25, 1981 and June 18, 1981. AMS (MOS) subject classifications (1970). Primary 20M10.

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The structure of pseudo-inverse semigroups

Let S be a regular semigroup, and let a ∈ S . Then a variant of S with respect to a is a semigroup with underlying set S and multiplication \circ defined by x \circ y = xay . In this paper, we characterise the regularity preserving elements of regular semigroups; these are the elements a such that (S,\circ) is also regular. Hickey showed that the set of regularity preserving elements can function as a replacement for the unit group when S does not have an identity. As an application, we characterise the regularity preserving elements in certain Rees matrix semigroups. We also establish connections with work of Loganathan and Chandrasekaran, and with McAlister's work on inverse transversals in locally inverse semigroups. We also investigate the structure of arbitrary variants of regular semigroups concentrating on how the local structure of a semigroup affects the structure of its variants.

Variants of Regular Semigroups

We consider a Rees matrix semigroup S = M[U; I, J; P] over a semigroup U, with I and J finite index sets, and relate the automaticity of S with the automaticity of U. We prove that if U is an automatic semigroup and S is finitely generated then S is an automatic semigroup. If S is an automatic semigroup and there is an entry p in the matrix P such that pU 1 = U then U is automatic. We also prove that if S is a prefix-automatic semigroup, then U is a prefix-automatic semigroup.

On Automatic Rees Matrix Semigroups

Embedding any countable semigroup in a 2-generated bisimple monoid

A regular semigroup S is called an ℋ-coextension of a regular semigroup T if there exists an idempotent-separating homomorhism from S onto T. J. Meakin [5] has described all regular four-spiral semigroups, i.e. all ℋ-coextensions of the fundamental four-spiral semigroup Sp4 [2], by means of the structure mappings on a regular semigroup. The purpose of this note is to point out that D. Allen's generalization [1] of the Rees theorem allows one to give a short alternative description of all regular four-spiral semigroups and their maximum completely simple homomorphic images in terms of bisimple ω-semigroups (whose structure is known by Reilly's theorem [7]) and Rees matrix semigroups ℳ(S;I;ΛP) over a semigroup S [3]. The notion of a Rees matrix semigroup over a semigroup is also used to embed semigroups in idempotent-generated ones, providing easy proofs for some embedding theorems of F. Pastijn [6].

Regular four-spiral semigroups, idempotent-generated semigroups and the rees construction

Although the classes of regular simple semigroups and simple semigroups without idempotents are evidently at opposite ends of the spectrum of simple semigroups, their theories involve some interesting connections. Jones [5] has obtained analogues of the bicyclic semigroup for simple semigroups without idempotents. Megyesi and Pollak [7] have classified all combinatorial simple principal ideal semigroups on two generators, showing that all are homomorphic images of one such semigroup Po which has no idempotents.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/037DF69B1A110F16BE33FDE118C40589/S0017089500007126a.pdf/div-class-title-embedding-any-countable-semigroup-without-idempotents-in-a-2-generated-simple-semigroup-without-idempotents-div.pdf

Karl Byleen

Papers

Embedding any countable semigroup in a 2-generated bisimple monoid

Regular four-spiral semigroups, idempotent-generated semigroups and the rees construction

Embedding any countable semigroup without idempotents in a 2-generated simple semigroup without idempotents

Embedding any countable semigroup in a 2-generated congruence-free semigroup

Spirals of idempotents from P-semigroups