Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-16T14:49:57.898Z Has data issue: false hasContentIssue false
  • English
  • Français

Subdirect products of E–inversive semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

H. Mitsch
H. Mitsch
Affiliation:
University of ViennaA-1090 Wien, Austria
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A semigroup S is called E-inversive if for every a ∈ S ther is an x ∈ S such that (ax)2 = ax. A construction of all E-inversive subdirect products of two E-inversive semigroups is given using the concept of subhomomorphism introduced by McAlister and Reilly for inverse semigroups. As an application, E-unitary covers for an E-inversive semigroup are found, in particular for those whose maximum group homomorphic image is a given group. For this purpose, the explicit form of the least group congruence on an arbitrary E-inversive semigroup is given. The special case of full subdirect products of a semilattice and a group (that is, containing all indempotents of the direct product) is investigated and, following an idea of Petrich, a construction of all these semigroups is provided. Finally, all periodic semigroups which are subdirect products of a semilattice or a band with a group are characterized.

MSC classification

Secondary: 20M10: General structure theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 1 , February 1990 , pp. 66 - 78
Copyright
Copyright © Australian Mathematical Society 1990

References

Feigenbaum, R. (1979), ‘Congruences on regular semigroups’, Semigroup Forum 17, 373–377.CrossRefGoogle Scholar
Hall, T. E. and Munn, W. D. (1985), ‘The hypercore of a semigroup’, Proc. Edinburgh Math. Soc. 28, 107–112.CrossRefGoogle Scholar
Howie, J. and Lallement, G. (1965), ‘Certain fundamental congruences on a regular semigroup’, Glasgow Math. J. 7, 145–159.Google Scholar
Lallement, G. and Petrich, M. (1966), ‘Décompositions I-matricielles d'un demigroupe’, J. Math. Pures Appl. 45, 67–117.Google Scholar
Margolis, S. W. and Pin, J. E. (1987), ‘Inverse semigroups and extension of groups by semilattices’, J. Algebra 110, 277–298.CrossRefGoogle Scholar
McAlister, D. B. and Reilly, N. R. (1977), ‘E-unitary covers for inverse semigroups’, Pacific J. Math. 68 161–174.CrossRefGoogle Scholar
Mitsch, H. (1986), ‘A natural partial order for semigroups’, Proc. Amer. Math. Soc. 97, 384–388.CrossRefGoogle Scholar
Petrich, M. (1967), ‘Sur certaines classes de demigroups I, II’, Bull. Cl. Sci. Acad. Roy. Belgique 49, 785–798, 888–900.Google Scholar
Petrich, M. (1973a), Introduction to semigroups (Merill, Columbus, Ohio).Google Scholar
Petrich, M. (1973b), ‘Regular semigroups which are subdirect products of a band and a semilattice of groups’, Glasgow Math. J. 14, 27–49.CrossRefGoogle Scholar
Petrich, M. and Reilly, N. R. (1983), ‘E-unitary covers and varieties of inverse semigroups’, Acta Sci. Math. (Szeged) 46, 59–72.Google Scholar
Thierrin, G. (1955), ‘Demigroupes inversés et rectangularies’, Bull. Cl. Sci. Acad. Roy. Belgique 41, 83–92.Google Scholar