Semigroups with inverse skeletons and Zappa-Sz$\acute{\rm e}$p products

Document Type : Research Paper


Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom.


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 $ E\subseteq 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 ${\mathcal R},{\mathcal L},{\mathcal H}$ and $\mathcal D$ are replaced by $\widetilde{{\mathcal R}}_E,\widetilde{{\mathcal L}}_E, \widetilde{{\mathcal H}}_E$ 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 $E\subseteq U$ and $U$ intersects every $\widetilde{{\mathcal H}}_E$-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$\acute{\rm e}$p products. Our approach can be immediately applied to obtain corresponding results for bisimple inverse monoids.


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