Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T13:37:59.425Z Has data issue: false hasContentIssue false
  • English
  • Français

Congruence lattices of regular semigroups related to kernels and traces

Published online by Cambridge University Press:  20 January 2009

Mario Petrich
Rights & Permissions [Opens in a new window]

Extract

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 kernel–trace approach to congruences on a regular semigroup S can be refined by introducing the left and right traces. This induces eight operators on the lattice of congruences on S: t1, k, tr,; Tt, K, Tr; t, T. We describe the lattice of congruences on S generated by six 3-element subsets of the set {ωt1, ωk, ωtr, εTt, εK, εTr} where ω and ε denote the universal and the equality relations. This is effected by means of a diagram and in terms of generators and relations on a free distributive lattice, or a homomorphic image thereof. We perform the same analysis for the lattice of congruences on S generated by the set {εK, ωk, εT, ωt}.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 2 , June 1991 , pp. 179 - 203
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Grätzer, G., General Lattice Theory (Academic Press, New York, 1978).CrossRefGoogle Scholar
2.Howie, J. M., An Introduction to Semigroup Theory (Academic Press, London, 1976).Google Scholar
3.Pastijn, F., The lattice of completely regular semigroup varieties, preprint.Google Scholar
4.Pastijn, F. and Petrich, M., Congruences on a regular semigroup, Trans. Amer. Math. Soc. 295 (1986), 607–633.CrossRefGoogle Scholar
5.Pastijn, F. and Petrich, M., Congruences on regular semigroups and the Green relations, Boll. Un. Mat. Ital. IB (1987), 591–603.Google Scholar
6.Pastijn, F. and Trotter, P. G., Lattices of completely regular semigroup varieties, Pacific J. Math. 119 (1985), 191–214.CrossRefGoogle Scholar
7.Petrich, M., Regular semigroups which are subdirect products of a band and a semilattice of groups, Glasgow J. Math. 14 (1973), 27–49.CrossRefGoogle Scholar
8.Petrich, M., Lectures in Semigroups (Akademie-Verlag, Berlin, 1977).CrossRefGoogle Scholar
9.Petrich, M., The kernel relation for certain regular semigroups, preprint.Google Scholar
10.Petrich, M., The kernel relation for a retract extension of Brandt semigroups, preprint.Google Scholar
11.Petrich, M., A special sublattice of the congruence lattice of a regular semigroup, preprint.Google Scholar
12.Petrich, M. and Reilly, N. R., A network of congruences on an inverse semigroup, Trans. Amer. Math. Soc. 270 (1982), 309–325.CrossRefGoogle Scholar
13.Petrich, M. and Reilly, N. R., Semigroups generated by certain operators on varieties of completely regular semigroups, Pacific J. Math. 132 (1988), 151–175.CrossRefGoogle Scholar