Semigroups with inverse skeletons and Zappa-Szep 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 RE; LE; HE and DE. 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 HE-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 DE-simple monoid. Using these techniques, we show that a reasonably wide class of DE-simple monoids can be decomposed as Zappa-Szep products. Our approach can be immediately applied to obtain corresponding results for bisimple inverse monoids.