Gould, V., Zenab, R. (2013). Semigroups with inverse skeletons and Zappa-Sz'{e}p products. Categories and General Algebraic Structures with Applications, 1(1), 59-89.

Victoria Gould; Rida-e- Zenab. "Semigroups with inverse skeletons and Zappa-Sz'{e}p products". Categories and General Algebraic Structures with Applications, 1, 1, 2013, 59-89.

Gould, V., Zenab, R. (2013). 'Semigroups with inverse skeletons and Zappa-Sz'{e}p products', Categories and General Algebraic Structures with Applications, 1(1), pp. 59-89.

Gould, V., Zenab, R. Semigroups with inverse skeletons and Zappa-Sz'{e}p products. Categories and General Algebraic Structures with Applications, 2013; 1(1): 59-89.

Semigroups with inverse skeletons and Zappa-Sz'{e}p products

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

Abstract

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.

[1] G. Casadio, `Construzione di gruppi come prodotto di sottogruppi permutabili' Univ. Roma e Ist. Naz. Alta Mat. Rend. Mat. e Appl 5 (1941), 348{360. [2] C. Cornock, Restriction Semigroups: Structure, Varieties and Presentations PhD thesis, York, 2011. [3] D. Easdown, `Biordered sets come from semigroups', J. Algebra 96 (1985), 581-591. [4] J.B. Fountain, `Products of idempotent integer matrices', Math. Proc. Camb. Phil. Soc. 11 (1991), 431-441. [5] J. Fountain, G. M. S. Gomes and V. Gould, `The free ample monoid', I.J.A.C. 19 (2009), 527{554. [6] V. Gould, `Notes on restriction semigroups and related structures; http://wwwusers.york.ac.uk/varg1/restriction.pdf. [7] T.E. Hall, `Some properties of local subsemigroups inherited by larger subsemigroups', Semigroup Forum 25 (1982), 35-49. [8] J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, Oxford, 1995. [9] M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts, and Categories, de Gruyter, Berlin, 2000. [10] M. Kunze, `Zappa products', Acta Math. Hungarica 41 (1983), 225-239. [11] M. Kunze, `Bilateral semidirect products of transformation semigroups', Semigroup Forum 45 (1992), 166-182. [12] M. Kunze, `Standard automata and semidirect products of transformation semigroups', Theoret. Comput. Sci 108 (1993), 151-171. [13] T.G. Lavers, `Presentations of general products of monoids', J. Algebra 204 (1998), 733-741. [14] K.S.S. Nambooripad, `Structure of regular semigroups. I', Memoirs American Math. Soc. 224 (1979). [15] B.H. Neumann, `Decompositions of groups', J. London Math. Soc. 10 (1935), 36. [16] J. Szep, `On factorizable, not simple groups', Acta Univ. Szeged. Sect. Sci. Math 13 (1950), 239{241. [17] J. Szep, ` Uber eine neue Erweiterung von Ringen', Acta Sci. Math. Szeged 19 (1958), 51.

[18] J. Szep, `Sulle strutture fattorizzabili', Atti Accad. Naz. Lincei Rend. CI. sci. Fis. Mat. Nat 8 (1962), 649{652. [19] G. Zappa, `Sulla construzione dei gruppi prodotto di due sottogruppi permutabilitra loro', Atti Secondo Congresso Un. Ital. Bologana p.p. 119{125, 1940.