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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 Mathematik UniversitätWien Strudlhofgasse 4 A-1090 Wien, Austria
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Abstract

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In this note a characterization of semigroups with atomistic consruence lattices, given for weakly reductive semigroups, is generalized to arbitrary semigroups. Also, it is shown that there is a complete congruence on the congruence lattice of such a semigroup that decomposes it into a disjoint union of intervals of the partition lattice.

MSC classification

Secondary: 20M10: General structure theory 08A30: Subalgebras, congruence relations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 52 , Issue 1 , February 1992 , pp. 88 - 102
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Auinger, K., ‘Semigroups with complemented congruence lattices’, Algebra Universalis 22 (1986), 192–204.Google Scholar
[2]Auinger, K., ‘Weakly reductive semigroups with atomistic congruence lattices’, J. Austral. Math. Soc. (Series A) 49 (1990), 59–71.CrossRefGoogle Scholar
[3]Gluskin, L. M., ‘Simple semigroups with zero’, Dokl. Akad. Nauk. SSSR 103 (1955), 5–8 (Russian).Google Scholar
[4]Tully, E. J. Jr, ‘Semigroups in which each ideal is a retract’, J. Austral. Math. Soc. 9 (1969), 239–245.Google Scholar