Semigroups with Inverse Skeletons and Zappa-szép Products

The aim of this paper is to study semigroups possessing E-regular elements, where an element a of a semigroup S is E-regular if a has an inverse a◦ such that aa◦, a◦a lie in E ⊆ E(S). Where S possesses ‘enough’ (in a precisely defined way) E-regular elements, analogues of Green’s lemmas and even of Green’s theorem hold, where Green’s relations R,L,H and D are replaced by R̃E , L̃E , H̃E and D̃E . Note that S itself need not be regular. We also obtain results concerning the extension of (one-sided) congruences, which we apply to (one-sided) congruences on maximal subgroups of regular semigroups. If S has an inverse subsemigroup U of E-regular elements, such that E ⊆ U and U intersects every H̃E-class exactly once, then we say that U is an inverse skeleton of S. We give some natural examples of semigroups possessing inverse skeletons and examine a situation where we can build an inverse skeleton in a D̃E-simple monoid. Using these techniques, we show that a reasonably wide class of D̃E-simple monoids can be decomposed as Zappa-Szép products. Our approach can be immediately applied to obtain corresponding results for bisimple inverse monoids.