Semigroups with inverse skeletons and Zappa-Szep products

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.
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



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.

C. Cornock, Restriction Semigroups: Structure, Varieties and Presentations PhD

thesis, York, 2011.

D. Easdown, `Biordered sets come from semigroups', J. Algebra 96 (1985), 581{

J.B. Fountain, `Products of idempotent integer matrices', Math. Proc. Camb.

Phil. Soc. 11 (1991), 431{441.

J. Fountain, G. M. S. Gomes and V. Gould, `The free ample monoid', I.J.A.C.

(2009), 527{554.

V. Gould, `Notes on restriction semigroups and related structures; http://wwwusers.

T.E. Hall, `Some properties of local subsemigroups inherited by larger subsemigroups',

Semigroup Forum 25 (1982), 35{49.

J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, Oxford,

M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts, and Categories, de Gruyter,

Berlin, 2000.

M. Kunze, `Zappa products', Acta Math. Hungarica 41 (1983), 225{239.

M. Kunze, `Bilateral semidirect products of transformation semigroups', Semigroup

Forum 45 (1992), 166{182.

M. Kunze, `Standard automata and semidirect products of transformation semigroups',

Theoret. Comput. Sci 108 (1993), 151{171.

T.G. Lavers, `Presentations of general products of monoids', J. Algebra 204

(1998), 733{741.

K.S.S. Nambooripad, `Structure of regular semigroups. I', Memoirs American

Math. Soc. 224 (1979).

B.H. Neumann, `Decompositions of groups', J. London Math. Soc. 10 (1935),


Semigroups with inverse skeletons and Zappa-Szep products 89

J. Szep, `On factorizable, not simple groups', Acta Univ. Szeged. Sect. Sci. Math

(1950), 239{241.

J. Szep, ` Uber eine neue Erweiterung von Ringen', Acta Sci. Math. Szeged 19

(1958), 51{62.

J. Szep, `Sulle strutture fattorizzabili', Atti Accad. Naz. Lincei Rend. CI. sci.

Fis. Mat. Nat 8 (1962), 649{652.

G. Zappa, `Sulla construzione dei gruppi prodotto di due sottogruppi permutabili

tra loro', Atti Secondo Congresso Un. Ital. Bologana p.p. 119{125, 1940.

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