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

Document Type: Research Paper

Authors

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 $ 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.

Keywords

References

[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.

Categories and General Algebraic Structures with Applications

Volume 1, Issue 1
Summer and Autumn 2013
Pages 59-89

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