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A syntactic approach to covers for E-dense semigroups over group varieties

Part of: Semigroups

Published online by Cambridge University Press:  26 February 2010

K. Auinger  and
P. G. Trotter
K. Auinger
Affiliation:
Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria. E-mail: karl.auinger@univie.ac.at
P. G. Trotter
Affiliation:
School of Mathematics and Physics, University of Tasmania, 7001 Hobart, Tasmania, Australia, E-mail: trotter@hilbert.maths.utas.edu.au
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Abstract

A major result of D. B. McAlister is that every inverse semigroup is an idempotent separating morphic image of an E-unitary inverse semigroup. The result has been generalized by various authors (including Szendrei, Takizawa, Trotter, Fountain, Almeida, Pin, Weil) to any semigroup of the following types: orthodox, regular, ii-dense with commuting idempotents, E-dense with idempotents forming a subsemigroup, and is-dense. In each case, a semigroup is a morphic image of a semigroup in which the weakly self conjugate core is unitary and separated by the homomorphism. In the present paper, for any variety H of groups and any E-dense semigroup S, the concept of an “H-verbal subsemigroup” of S is introduced which is intimately connected with the least H-congruence on S. What is more, this construction provides a short and easy access to covering results of the aforementioned kind. Moreover, the results are generalized, in that covers over arbitrary group varieties are constructed for any E-dense semigroup. If the given semigroup enjoys a “regularity condition” such as being eventually regular, group bound, or regular, then so does the cover.

MSC classification

Secondary: 20M10: General structure theory
Type
Research Article
Information
Mathematika , Volume 48 , Issue 1-2 , December 2001 , pp. 231 - 245
Copyright
Copyright © University College London 2001

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