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 $a^circ$ such that $aa^circ,a^circ a$ lie in $ esubseteq 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 $mbox{$mathcal r$},el,eh$ and $dee$ are replaced by $art_e,elt_e, eht_e$ and $widetilde{mathcal{d}}_e$. 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 $esubseteq u$ and $u$ intersects every $eht_e$-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 $widetilde{mathcal{d}}_e$-simple monoid. using these techniques, we show that a reasonably wide class of $widetilde{mathcal{d}}_e$-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.