Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups

McAlister, D. B.

doi:10.1017/s1446788700019467

Cited by 51 publications

(53 citation statements)

References 9 publications

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“…However, the set of regular elements forms a (regular) subsemigroup. This crucial observation is due to McAlister [15]. The subsemigroup Reg(.M(/, 5, A; P)) is usually denoted UM(I, S,A;P) and is called the / x A regular Rees matrix semigroup over S with sandwich matrix P. Regular Rees matrix semigroups over inverse semigroups are of special importance (see [15,16] …”

Section: Preliminariesmentioning

confidence: 99%

Rees matrix semigroups and the regular semidirect product

Auinger

Szendrei

2005

J. Aust. Math. Soc.

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A generalization of the Pastijn product is introduced so that, on the level of e-varieties and pseudoe-varieties, this product and the regular semidirect product by completely simple semigroups 'almost always' coincide. This is applied to give a model of the bifree objects in every e-variety formed as a regular semidirect product of a variety of inverse semigroups by a variety of completely simple semigroups that is not a group variety.2000 Mathematics subject classification: primary 20M17, 20M07, 20M10.

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Section: Preliminariesmentioning

confidence: 99%

Rees matrix semigroups and the regular semidirect product

Auinger

Szendrei

2005

J. Aust. Math. Soc.

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“…McAlister [23] showed that if S is a regular semigroup then the regular elements of M(I, S, Λ, P ) form a subsemigroup, which he called a regular Rees matrix semigroup and denoted RM(I, S, Λ, P ). We prefer the notation Reg M(I, S, Λ, P ).…”

Section: Semidirect Products Of Regular Semigroups 4283mentioning

confidence: 99%

Untitled

1997

Trans. Amer. Math. Soc.

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Abstract. Within the usual semidirect product S * T of regular semigroups S and T lies the set Reg (S * T ) of its regular elements. Whenever S or T is completely simple, Reg (S * T ) is a (regular) subsemigroup. It is this 'product' that is the theme of the paper. It is best studied within the framework of existence (or e-) varieties of regular semigroups. Given two such classes, U and V, the e-variety U * V generated by {Reg (S * T ) : S ∈ U, T ∈ V} is well defined if and only if either U or V is contained within the e-variety CS of completely simple semigroups. General properties of this product, together with decompositions of many important e-varieties, are obtained. For instance, as special cases of general results the e-variety LI of locally inverse semigroups is decomposed as I * RZ, where I is the variety of inverse semigroups and RZ is that of right zero semigroups; and the e-variety ES of E-solid semigroups is decomposed as CR * G, where CR is the variety of completely regular semigroups and G is the variety of groups.In the second half of the paper, a general construction is given for the e-free semigroups (the analogues of free semigroups in this context) in a wide class of semidirect products U * V of the above type, as a semidirect product of e-free semigroups from U and V, "cut down to regular generators". Included as special cases are the e-free semigroups in almost all the known important e-varieties, together with a host of new instances. For example, the e-free locally inverse semigroups, E-solid semigroups, orthodox semigroups and inverse semigroups are included, as are the e-free semigroups in such sub-e-varieties as strict regular semigroups, E-solid semigroups for which the subgroups of its self-conjugate core lie in some given group variety, and certain important varieties of completely regular semigroups. Graphical techniques play an important role, both in obtaining decompositions and in refining the descriptions of the e-free semigroups in some e-varieties.Similar techniques are also applied to describe the e-free semigroups in a different 'semidirect' product of e-varieties, recently introduced by Auinger and Polák. The two products are then compared.The semidirect product has a venerable history in semigroup theory. In conjunction with the wreath product, it has played a central role in the decomposition theory of finite semigroups. In the context of regular semigroups, variations on the usual product have been introduced for inverse semigroups and, recently, for locally inverse semigroups. We take a different approach, by studying the set Reg (S * T ) of regular elements of the usual semidirect product of regular semigroups S and T . In many important cases, these elements form a (regular) subsemigroup of S * T . These cases are most easily interpreted in the framework of e-varieties.

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“…To show that S contains a subsemigroup isomorphic to Sp 4 it suffices by Lemma 3.1 to show that 91L does. The maximum idempotent-separating congruence n on eSe induces an idempotent-separating homomorphism from L(eSe; m, m; P) into 91L(e5e/ju; m, m; P^) where for P = (/?, 7 ) we denote by Pl^ the matrix (p^fi*). Let £ = £ eS< ,.…”

Section: Let S Be a Non-completely Simple Bisimple Semigroup Which Ismentioning

confidence: 99%

On bisimple semigroups generated by a finite number of idempotents

Byleen

1982

J Aust Math Soc A

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Non-completely simple bisimple semigroups 5 which are generated by a finite number of idempotents are studied by means of Rees matrix semigroups over local submonoids eSe, e = e 2 G S. If under the natural partial order on the set £ ( of idempotents of such a semigroup S the sets u(e) = {/ G E s : f =s e) for each e G E s are well-ordered, then S is shown to contain a subsemigroup isomorphic to Sp 4 , the fundamental four-spiral semigroup. A non-completely simple bisimple semigroup is constructed which is generated by 5 idempotents but which does not contain a subsemigroup isomorphic to S/) 4 .

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Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups

Cited by 51 publications

References 9 publications

Rees matrix semigroups and the regular semidirect product

Rees matrix semigroups and the regular semidirect product

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On bisimple semigroups generated by a finite number of idempotents

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