# E-Inversive Γ-Semigroups

30 Sep 2009-Kyungpook Mathematical Journal (Department of Mathematics, Kyungpook National University)-Vol. 49, Iss: 3, pp 457-471

TL;DR: In this article, the authors investigate different properties of E-inversive -semigroup and right E-semigroup, and show that a right E--semigroup is a -idempotent.

Abstract: Let S = {a, b, c, ...} and = {, , , ...} be two nonempty sets. S is called a -semigroup if , for all and a, b S and , for all a, b, c S and for all , . An element is said to be an -idempotent for some if = e. A -semigroup S is called an E-inversive -semigroup if for each there exist and such that ax is a -idempotent for some . A -semigroup is called a right E--semigroup if for each -idempotent e and -idempotent f, is a -idempotent. In this paper we investigate different properties of E-inversive -semigroup and right E--semigroup.

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01 Jan 2016TL;DR: A survey of some works published by different authors on the concept of gamma-semigroups can be found in this article, where the authors present a survey of the generalization of semigroups.

Abstract: The concept of \(\Gamma \)-semigroup is a generalization of semigroup. Let S and \(\Gamma \) be two nonempty sets. S is called \(\Gamma \)-semigroup if there exists a mapping \(S\times \Gamma \times S\longrightarrow S\), written as \((a, \alpha , b)\longrightarrow a\alpha b\), satisfying the identity \( (a\alpha b)\beta c\) \(=\) \(a\alpha (b\beta c) \) for all \(a, b, c\in S\) and \( \alpha , \beta \in \Gamma \). This article is a survey of some works published by different authors on \(\Gamma \)-semigroups.

2 citations

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TL;DR: In this paper, a semigroup S is called E-(0-) inversive if for every a ∈ S4(a ≠ 0), there exists x ∈ s such that ax (≠ 0) is an idempotent.

Abstract: Generalizing a property of regular resp. finite semigroups a semigroup S is called E-(0-) inversive if for every a ∈ S4(a ≠ 0) there exists x ∈ S such that ax (≠ 0) is an idempotent. Several characterizations are given allowing to identify the (completely, resp. eventually) regular semigroups in this class. The case that for every a ∈ S4(≠ 0) there exist x,y ∈ S such that ax = ya(≠ 0) is an idempotent, is dealt with also. Ideal extensions of E- (0-)inversive semigroups are studied discribing in particular retract extensions of completely simple semigroups. The structure of E- (0-)inversive semigroups satisfying different cancellativity conditions is elucidated. 1991 AMS classification number: 20M10.

23 citations

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TL;DR: In this paper, alternative characterizations of semidirect products of semigroups are given, such as groups, regular semiggroups, and inverse semiigroups. But they do not consider the relation between groups and regular semigenes.

Abstract: Two alternative characterizations of semidirect products of semigroups are given. Characterizations are provided of such products that are groups, regular semigroups, and inverse semigroups, respectively.

22 citations

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01 Jan 1989

TL;DR: In this paper, the authors determine which semidirect products of semigroups are E-inversive and E-dense, respectively, and show that the case in which S induces only automorphism on T allows a particularly simple description.

Abstract: Let X be a subset of a semigroup S We denote by E(X) the set of idempotent elements Of XAn element a of a semigroup S is called E-inverse if there exists such that We note that the definition is not one-sided Indeed, an a element of a semigroup S is E-inversive if there exists such that (see [7], [l] p 98) A semigroup S is called E-inversive if all its elements are E-inversive This class of semigroups is extensive All semigroups with a zero and all eventually regular semigroups [2] are E-inversive semigroupsRecently E-inversive semigroups reappeared in a paper by Hall and Munn [3] and in a paper by Mitsch [5] The special case of E-inversive semigroups with pairwise commuting idempotents, called E-dense, was considered by Margolis and Pin [4] Let S and T be semigroups, and let be a homomorphism of S into the endomorphism semigroup of T If and , denote Thus, if and then The semidirect product of S and T ,in that order, with strutture map (Y, consists of the set S x T equipped with the product This product will be denoted by S _{𝛼}T In this note we determine which semidirect products of semigroups are E-inversive semigroups and E-dense semigroups, respectively It turns out that the case in which S induces only automorphism on T allows a particularly simple description In [6], Preston has answered the analous question for regular semigroups and for inverse semigroups For the terminology and for the definitions of the algebraic theory of semigroups, we refer to [1]

14 citations

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Abstract: A semigroup S is called E-inversive if for every a ∈ S there is an x ∈ S such that ax is idempotent. The purpose of this paper is the investigation of E-inversive semigroups and semigroups whose idempotents form a subsemigroup. Basic properties are analysed and, in particular, semigroups whose idempotents form a semilattice or a rectangular band are considered. To provide examples and characterizations, the construction methods of generalized Rees matrix semigroups and semidirect products are employed.

9 citations

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TL;DR: In this article, the semidirect product of a semigroup and a Γ-semigroup is studied and some interesting properties of this product are investigated. And the notion of wreath product is introduced.

Abstract: Let S = {a, b, c, . . .} and Γ = {α, β, γ, . . . } be two nonempty sets. S is called a Γ-semigroup if aαb ∈ S, for all α ∈ Γ and a, b ∈ S and (aαb)βc = aα(bβc), for all a, b, c ∈ S and for all α, β ∈ Γ. In this paper we study the semidirect product of a semigroup and a Γ-semigroup. We also introduce the notion of wreath product of a semigroup and a Γsemigroup and investigate some interesting properties of this product.

3 citations