Semigroups with inverse skeletons and Zappa-Szep products

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Abstract


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.

Keywords


idempotents, R;L, restriction semigroups, Zappa-Szep products. Subject Classication[2010]: 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

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