@MISC{Billhardt_ageneralization, author = {Bernd Billhardt}, title = {A Generalization of F-regular Semigroups}, year = {} }

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Abstract

A regular semigroup S is termed locally F-regular, if each class of the least completely simple congruence ξ contains a greatest element with respect to the natural partial order. It is shown that each locally F-regular semigroup S admits an embedding into a semidirect product of a band by S/ξ. Further, if ξ satisfies the additional property that for each s ∈ S and each inverse (sξ) ′ of sξ in S/ξ the set (sξ) ′ ∩ V (s) is not empty, we represent S both as a Rees matrix semigroup over an F-regular semigroup as well as a certain subsemigroup of a restricted semidirect product of a band by S/ξ. The main result contains our recent representation theorem for F-regular semigroups [1], whence McAlister’s characterization of F-inverse semigroups [2]. Finally, we establish that an orthodox semigroup S is a homomorphic image of an F-regular semigroup, if and only if it contains an inverse subsemigroup S 0 such that (i) for each s ∈ S there is s 0 ∈ S 0 with ss 0 s = s, (ii) the idempotents of S 0 commute with the idempotents of S. In particular, we recapture a result due to McFadden, which states that each unit-regular orthodox semigroup is an idempotent separating homomorphic image of a semidirect product of a band with identity by a group [3].