Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-24T05:12:20.675Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Almeida, J., On hyperdecidable pseudovarieties of simple semigroups. Internal. J. Algebra Comput. 10 (2000), 261–284.CrossRefGoogle Scholar
2.Almeida, J. and Escada, A., On the equation V*G = V. J. Pure Appl. Algebra 166 (2002), 1–28.CrossRefGoogle Scholar
3.Almeida, J.Pin, J.-E. and Weil, P., Semigroups whose idempotents form a subsemigroup. Math. Proc. Camb. Phil. Soc. 111 (1992), 241–253.CrossRefGoogle Scholar
4.Ash, C. J., Inevitable graphs: a proof of the type II conjecture and some related decision procedures. Internat. J. Algebra Comput. 1 (1991), 127–146.CrossRefGoogle Scholar
5.Edwards, P. M., Eventually regular semigroups. Bull. Austral. Math. Soc. 28 (1983), 23–38.CrossRefGoogle Scholar
6.Fountain, J., E-unitary covers of E-dense monoids. Bull. London Math. Soc. 22 (1990), 353–358.CrossRefGoogle Scholar
1.Fountain, J.Pin, J.-E. and Weil, P., Covers for monoids. J. Algebra (to appear).Google Scholar
8.Howie, J. M., Fundamentals of Semigroup Theory, (Clarendon Press, Oxford, 1995).CrossRefGoogle Scholar
9.Gomes, G. M. S., A characterization of the group congruences on a semigroup. Semigroup Forum 46 (1993), 48–53.CrossRefGoogle Scholar
10.Gomes, G. M. S. and Gould, V., Proper weakly left ample semigroups. Internat. J. Algebra Comput. 9 (1999), 721–739.CrossRefGoogle Scholar
11.LaTorre, D. R., Group congruences on regular semigroups. Semigroup Forum 24 (1982), 327–340.CrossRefGoogle Scholar
12.McAlister, D. B., Groups, semilattices and inverse semigroups. Trans. Amer. Math. Soc. 192 (1974), 237–244.Google Scholar
13.McAlister, D. B. and Reilly, N. R., E-unitary covers for inverse semigroups. Pacific J. Math. 68(1977), 161–174.CrossRefGoogle Scholar
14.Mitsch, H., Subdirect products of E-inversive semigroups. J. Austral. Math. Soc. 48 (1990), 66–78.CrossRefGoogle Scholar
15.Mitsch, H., Introduction to E-inversive semigroups. In: Semigroups (Smith, P.Giraldes, E.Martins, P., eds.) (World Scientific: Singapore, 2000), pp. 114–135.CrossRefGoogle Scholar
16.Neumann, H., Varieties of Groups (Springer-Verlag, Berlin Heidelberg, New York, 1967).CrossRefGoogle Scholar
17.Petrich, M. and Reilly, N. R., E-unitary covers and varieties of inverse semigroups. Acta. Sci. Math. Szeged 46 (1983), 59–72.Google Scholar
18.Szendrei, M. B., On a pull back diagram for orthodox semigroups. Semigroup Forum 20 (1980), 1–10. Corrigendum, 25 (1982), 311-324.CrossRefGoogle Scholar
19.Takizawa, K., Orthodox semigroups and E-unitary regular semigroups. Bull. Tokyo Gakugei University (4) 31 (1979), 41–43.Google Scholar
20.Trotter, P. G., Covers for regular semigroups and an application to complexity. J. Pure Appl. Algebra 105 (1995), 319–328.CrossRefGoogle Scholar
21.Trotter, P. G. and Jiang, Z., Covers for regular semigroups. SEA Bull. Math. 18 (1994), 167–161.Google Scholar
22.Weipoltshammer, B., Kongruenzen auf E-inversiven E-Halbgruppen (Dissertation, Universitat Wien, 2000).Google Scholar