Rees Matrix Covers for Locally Inverse Semigroups

McAlister, D. B.

doi:10.2307/1999233

Cited by 21 publications

(56 citation statements)

References 3 publications

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“…Proof, (i) Since S is a quasi-ideal of T, every idempotent separating congruence a on S generates an idempotent separating congruence on T extending a by Proposition 2.4 of [27]. Uniqueness follows from Proposition 5.2(ii).…”

Section: Ii) If a Is An Idempotent Separating Congruence On S Then Imentioning

confidence: 99%

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Enlargements of regular semigroups

Lawson

1996

Proceedings of the Edinburgh Mathematical Society

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Section: Ii) If a Is An Idempotent Separating Congruence On S Then Imentioning

confidence: 99%

“…It was proved as Proposition 1.4 of [27]. As usual, y" denotes the natural map associated with the congruence y.…”

Section: Ii) Let X Ye S X € V(x) Y E V{y) and H E S(x'x Yy) Thenmentioning

confidence: 99%

Enlargements of regular semigroups

Lawson

1996

Proceedings of the Edinburgh Mathematical Society

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“…A regular semigroup S is said to be locally inverse if eSe is inverse for each idempotent e in S. Division theorems for locally inverse semigroups have been given by F. J. Pastijn [16] and the author [10]. In this section we shall use the use, available at https://www.cambridge.org/core/terms.…”

Section: Locally Inverse Semigroupsmentioning

confidence: 99%

Some covering theorems for locally inverse semigroups

McAlister

1985

J Aust Math Soc A

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A regular semigroup 5 is said to be locally inverse if each local submonoid eSe, with e an idempotent, is an inverse semigroup. In this paper we apply known covering theorems for inverse semigroups and a covering theorem for locally inverse semigroups due to the author to obtain some covering theorems for locally inverse semigroups. The techniques developed here permit us to give an alternative proof for, and slight strengthening of, an important covering theorem for locally inverse semigroups due to F. Pastijn.

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“…Now we state the theorem of McAlister to be used in the sequel, in its most general form [25] (originally proved for locally inverse semigroups in [24]). A morphism S → T of regular semigroups is a local isomorphism if it is injective on the local monoids of S. McAlister proves, as well, that the result is true for U the e-variety of L-unipotent regular semigroups (and its dual), but we shall concentrate on the cases listed.…”

Section: Corollary 43 Let U Be An E-variety That Is Not Contained Imentioning

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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Rees Matrix Covers for Locally Inverse Semigroups

Cited by 21 publications

References 3 publications

Enlargements of regular semigroups

Enlargements of regular semigroups

Some covering theorems for locally inverse semigroups

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