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.