On inverses of products of idempotents in regular semigroups

FitzGerald, D. G.

doi:10.1017/s1446788700013756

Cited by 78 publications

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“…1 Clearly, IG(E) is idempotent generated, and there is a natural map φ : IG(E) → S, given byēφ = e, such that im φ = S = E . Further, we have the following result, taken from [2,13,15,21,29], which exhibits the close relation between the structure of the regular D-classes of IG(E) and those of S.…”

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confidence: 55%

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Free idempotent generated semigroups and endomorphism monoids of independence algebras

Yang

Gould

2016

Semigroup Forum

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We study maximal subgroups of the free idempotent generated semigroup IG(E), where E is the biordered set of idempotents of the endomorphism monoid End A of an independence algebra A, in the case where A has no constants and has finite rank n. It is shown that when n ≥ 3 the maximal subgroup of IG(E) containing a rank 1 idempotent ε is isomorphic to the corresponding maximal subgroup of End A containing ε. The latter is the group of all unary term operations of A. Note that the class of independence algebras with no constants includes sets, free group acts and affine algebras.

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confidence: 55%

“…Exactly as in [19], we may use results of [23], [15] and [5] to locate a set of generators for H = H 11 . Note that the assumption that the set of generators in [5] is finite is not critical.…”

Section: = M(h ; I ; P)mentioning

confidence: 99%

Free idempotent generated semigroups and endomorphism monoids of independence algebras

Yang

Gould

2016

Semigroup Forum

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“…Proof, (i) The subsemigroup generated by the idempotents of a regular semigroup is regular (Fitz-Gerald [3]) and then from Result 1 and its dual it is easily seen that the subsemigroup generated by the idempotents of a union of groups is also a union of groups.…”

Section: Corollarymentioning

confidence: 99%

Amalgamation and inverse and regular semigroups

Hall¹

1978

Trans. Amer. Math. Soc.

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Abstract. A method for proving the embeddability of semigroup amalgams is introduced. After providing necessary and sufficient conditions in terms of representations for the weak embeddability of a semigroup amalgam, it successfully deals with the embedding of inverse semigroup amalgams into inverse semigroups and the embedding of an amalgam of regular semigroups whose core is full in each member.Most of the known results on amalgamation of semigroups, almost all of which are due to J. M. Howie, have now been proved by a method introduced in [7] involving a countably infinite number of steps, each step extending a representation of a semigroup [7], [8], [14], [15]; the main results thus proved concern amalgamation over a common unitary subsemigroup, an almost unitary subsemigroup, an inverse subsemigroup, a two-element subsemigroup, and the embedding of an inverse semigroup amalgam in an inverse semigroup. We give here a method of proof that avoids this infinite number of steps.First it yields a necessary and sufficient condition, in terms of representations, for the weak embeddability of a semigroup amalgam (defined below).This gives then a short proof of [7, Theorem 8] (see also Howie's text [9] for an exposition), namely that inverse semigroups have the strong amalgamation property. Further we are able to show that finitehess can be preserved in the embedding of an amalgam (S¡, i E I; U) of inverse semigroups if the common inverse subsemigroup U is full in each S¡, i.e. contains all the idempotents of each S¡; in general, finiteness cannot be preserved [7, §3].Further we are able to show that, in fact, regular semigroups can be amalgamated over a full regular subsemigroup, also with finiteness being preserved. In general, an amalgam of regular semigroups (even left regular bands) cannot be weakly embedded in a semigroup [8, Remark 7]. In the final section we consider semigroups that are unions of groups.

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“…Let a; b be elements of a semigroup S . [14]). Let S be a regular semigroup with set E of idempotents.…”

Section: Introductionmentioning

confidence: 99%

On Vⁿ-semigroups

Tang

2015

Open Mathematics

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Abstract:In this paper, we give some new characterizations of orthodox semigroups in terms of the set of inverses of idempotents. As a generalization, a new class of regular semigroups, namely V n -semigroups, is introduced. Also, we give a characterization of V n -semigroups and investigate some properties of V n -semigroups. Furthermore, weshow that the class of V n -semigroups is closed under direct products and homomorphic images. However, regular subsemigroups of V n -semigroups (n 2) are not necessarily V n -semigroups in general. Therefore, the class of V n -semigroups (n 2) does not form an e-variety. Finally, we obtain that a E-solid semigroup S is a V 2 -semigroup if and only if S is orthodox.

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On inverses of products of idempotents in regular semigroups

Cited by 78 publications

References 3 publications

Free idempotent generated semigroups and endomorphism monoids of independence algebras

Free idempotent generated semigroups and endomorphism monoids of independence algebras

Amalgamation and inverse and regular semigroups

On Vⁿ-semigroups

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On inverses of products of idempotents in regular semigroups

Cited by 78 publications

References 3 publications

Free idempotent generated semigroups and endomorphism monoids of independence algebras

Free idempotent generated semigroups and endomorphism monoids of independence algebras

Amalgamation and inverse and regular semigroups

On Vn-semigroups

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On Vⁿ-semigroups