Abstract. 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; aa lie in E E(S). Where S possesses `enough' (in a precisely dened 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 eRE; eLE; eHE and eDE. 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 eHE-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 eDE-simple monoid. Using these techniques, we showthat a reasonably wide class of eDE-simple monoids can be decomposed as Zappa-Szep products. Our approach can be immediately applied to obtain corresponding results for bisimple inverse monoids.
idempotents, R;L, restriction semigroups, Zappa-Szep products. Subject Classication: 20M10. The second author is grateful to the Schlumberger Foundation for funding her Ph.D. studies, of which this paper forms a part. The authors would also like t