Semigroups with inverse skeletons and Zappa-Sz'{e}p products

The aim of this paper is to study semigroups possessing E-regular elements, where an element a of a semigroup S is {em E-regular} if a has an inverse acirc such that aacirc,acirca lie in EsubseteqE(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 mbox$mathcalR$,el,eh and dee are replaced by artE,eltE,ehtE and widetildemathcalDE. 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 EsubseteqU and U intersects every ehtE-class exactly once, then we say that U is an {em 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 widetildemathcalDE-simple monoid. Using these techniques, we show that a reasonably wide class of widetildemathcalDE -simple monoids can be decomposed as Zappa-Sz'{e}p products. Our approach can be immediately applied to obtain corresponding results for bisimple inverse monoids.