On semidirect product of semigroups

TLDR: 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]