Hostname: page-component-84b7d79bbc-x5cpj Total loading time: 0 Render date: 2024-07-28T01:51:08.499Z Has data issue: false hasContentIssue false
  • English
  • Français

On the lattice of congruences on a semilattice

Published online by Cambridge University Press:  09 April 2009

T. E. Hall
T. E. Hall
Affiliation:
Monash UniversityClayton
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.

Lattices of congruences are studied in section II.6 of Cohn [2]. Papers [5] and [6] by Munn deal with lattices of congruences on semigroups and conditions under which these lattices are modular. In [4] Lallement shows that the lattice of congruences on a completely 0-simple semigroup is semimodular, giving an alternative proof of the result, due to Preston [7], that the lattice of congruences on a completely 0-simple semigroup satisfies a certain chain condition which is a natural extension to arbitrary lattices of the Jordan-Dedekind chain condition for finite lattices.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 4 , November 1971 , pp. 456 - 460
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Vol. I (Math. Surveys, number 7, Amer. Math. Soc., 1961).Google Scholar
[2]Cohn, P. M., Universal Algebra (Harper and Row, New York, 1965).Google Scholar
[3]Dubreil-Jacotin, M. L., Lesieur, L. and Croisot, R., Leçons sur Ia théorie des treillis des structures algébriques ordonnées et des treillis géométriques (Éditeur-imprimeur-libraire, GauthierVillars, Paris, 1953).Google Scholar
[4]Lallement, G., ‘Demi-groupes reguliers’, (Doctoral dissertation) Annali di Matematica pura ed applicata 77 (1967), 47–130.CrossRefGoogle Scholar
[5]Munn, W. D., ‘A certain sublattice of the lattice of congruences on a regular semigroup’, Proc. Camb. Phil. Soc. 60 (1964), 385–391.CrossRefGoogle Scholar
[6]Munn, W. D., ‘The lattice of congruences on a bisimple ω-semigroup’. Proc. Roy. Soc. Edinburgh, 67 (1966), 175–184.Google Scholar
[7]Preston, G. B., ‘Chains of congruences on a completely O-simple semigroup’, J. Aust. Math. Soc. 5 (1965), 76–82.CrossRefGoogle Scholar
[8]Reilly, N. R. and Scheiblich, H. E., ‘Congruences on regular semigroups’, Pacific J. Math. 23 (1967), 349–360.CrossRefGoogle Scholar