Hostname: page-component-669899f699-7tmb6 Total loading time: 0 Render date: 2025-04-26T20:27:57.471Z Has data issue: false hasContentIssue false
  • English
  • Français

Weakly reductive semigroups with atomistic congruence lattices

Part of: Semigroups Algebraic structures

Published online by Cambridge University Press:  09 April 2009

Karl Auinger
Karl Auinger
Affiliation:
Institut für MathematikUniversität WienA-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.

The structure of semigroups with atomistic congruence lattices (that is, each congruence is the supremum of the atoms it contains) is studied. For the weakly reductive case the problem of describing the structure of such semigroups is solved up to simple and congruence free semigroups, respectively. As applications, all commutative, finite, completely semisimple semigroups, respectively, with atomistic congruence lattices are described.

MSC classification

Secondary: 20M10: General structure theory 08A30: Subalgebras, congruence relations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 1 , August 1990 , pp. 59 - 71
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Auinger, K., ‘Semigroups with complemented congruence lattices’, Algebra Universalis 22 (1986), 192–204.CrossRefGoogle Scholar
[2]Auinger, K., ‘Semigroups with Boolean congruence lattices’, Monatsh. Math. 104 (1987), 1–15.Google Scholar
[3]Auinger, K., ‘Atomistic congruence lattices of semilattices’, Semigroup Forum, to appear.Google Scholar
[4]Clifford, A. H. and Preston, G. B. ‘The algebraic theory of semigroups,’ Vol. i, Amer. Math. Soc. Math. Surveys No. 7, Providence, R. I., 1961).Google Scholar
[5]Gluskin, L. M., ‘Simple semigroups with zero’, Dokl. Akad. Nauk SSSR 103 (1955), 5–8. (Russian)Google Scholar
[6]Howie, J. M., An introduction to semigroup theory, (Academic Press, New York, 1976).Google Scholar
[7]Lallement, G. and Petrich, M., ‘Structure d'sune classe de demi-groupes reguliers’, J. Math. Pures Appl. 48 (1969), 345–397.Google Scholar
[8]Petrich, M., Introduction to semigroups, (Merrill, Columbus, 1973).Google Scholar
[9]Petrich, M., ‘Structure of regular semigroups’ Cahiers Math. Montpellier 11, (Université du Langudoc, Montpellier, 1977).Google Scholar
[10]Schein, B. M., ‘Homomorphisms and subdirect decompositions of semigroups’, Pacific J. Math. 17 (1966), 529–547.CrossRefGoogle Scholar
[11]Tamura, T., ‘Indecomposable completely simple semigroups except groups’, Osaka Math. J. 8 (1956), 35–42.Google Scholar
[12]Tully, E. J. Jr, ‘Semigroups in which each ideal is a retract’, J. Austral. Math. Soc. 9 (1969), 239–245.CrossRefGoogle Scholar