On semidirect product of semigroups


Let X be a subset of a semigroup S. We denote by E(X) the set of idempotent elements Of X.An element a of a semigroup S is called E-inverse if there exists x ∈ E(S) such that ax ∈ E(S). We note that the definition is not one-sided. Indeed, an a element of a semigroup S is E-inversive if there exists y ∈ S such that ay, ya ∈ E(S) (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 semigroups.Recently 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 𝛼 : S → End(T) be a homomorphism of S into the endomorphism semigroup of T. If s ∈ S and t ∈ T, denote t(sa) by t<sup>s</sup>. Thus, if s,s' ∈ S and t ∈ T then {t<sup>s</sup>}<sup>s</sup> = t<sup>ss'</sup>. 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 (s,t)(s',t') = (ss', {t<sup>s'</sup>}{t'}). 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].

DOI Code: 10.1285/i15900932v9n2p189

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