Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-21T11:21:03.760Z Has data issue: false hasContentIssue false
  • English
  • Français

The lattice of completely regular semigroup varieties

Part of: Varieties Semigroups

Published online by Cambridge University Press:  09 April 2009

Francis Pastijn
Francis Pastijn
Affiliation:
Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, Wisconsin 53233, U.S.A.
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 completely regular semigroup is a semigroup which is a union of groups. The class CR of completely regular semigroups forms a variety. On the lattice L (CR) of completely regular semigroup varieties we define two closure operations which induce complete congruences. The consideration of a third complete congruence on L (CR) yields a subdirect decomposition of L (CR). Using these results we show that L (CR) is arguesian. This confirms the (tacit) conjecture that L (CR) is modular.

MSC classification

Secondary: 20M05: Free semigroups, generators and relations, word problems 20M07: Varieties and pseudovarieties of semigroups 08B15: Lattices of varieties
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 1 , August 1990 , pp. 24 - 42
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Birjukov, A. P., ‘Varieties of idempotent semigroups’, Algebra i Logika 9 (1970), 255–273. (in Russian).Google Scholar
[2]Clifford, A. H., ‘The free completely regular semigroup on a set’, J. Algebra 59 (1979), 434–451.CrossRefGoogle Scholar
[3]Fennemore, C. F., ‘All varieties of bands I’, Math. Nachr. 48 (1971), 237–252,CrossRefGoogle Scholar
II, Math. Nachr. 48 (1971), 253–262.Google Scholar
[4]Gerhard, J. A., ‘The lattice of equational classes of idempotent semigroups’, J. Algebra 15 (1979), 195–224.CrossRefGoogle Scholar
[5]Gerhard, J. A. and Petrich, M., ‘All varieties of regular orthogroups’, Semigroup Forum 31 (1985), 311–351.CrossRefGoogle Scholar
[6]Grätzer, G., General lattice theory, Birkhaüser, Basel, 1978.CrossRefGoogle Scholar
[7]Hall, T. E. and Jones, P. R., ‘On the lattice of varieties of bands of groups’, Pacific J. Math. 91 (1980), 327–337.CrossRefGoogle Scholar
[8]Howie, J. M., An introduction to semigroup theory, Academic Press, London, 1976.Google Scholar
[9]Jones, P. R., ‘Mal cev products of varieties of completely regular semigroups’, J. Austral Math. Soc. (Series A) 42 (1987), 227–246.CrossRefGoogle Scholar
[10]Pastijn, F., Congruences on regular semigroups—A survey, in Proceedings of the 1984 Marquette Conference of Semigroups, (Byleen, K., Jones, P., Pastijn, F., eds.), Marquette University, 1985, pp. 159–175.Google Scholar
[11]Pastijn, F. and Petrich, M., ‘Congruences on regular semigroups’, Trans. Amer. Math. Soc. 295 (1986), 607–633.CrossRefGoogle Scholar
[12]Pastijn, F. and Petrich, M., ‘The congruence lattice of a regular sermigroup’, J. Pure Appl. Algebra 53 (1988), 93–123.CrossRefGoogle Scholar
[13]Pastijn, F. and Trotter, P. G., ‘Lattices of completely regular semigroup varieties’, Pacific J. Math 119 (1985), 191–214.CrossRefGoogle Scholar
[14]Petrich, M., Lectures in semigroups, Wiley, London, 1977.CrossRefGoogle Scholar
[15]Petrich, M., ‘The structure of completely regular semigroups’, Trans. Amer. Math. Soc. 189 (1974), 211–236.CrossRefGoogle Scholar
[16]Polák, L., ‘On varieties of completely regular semigroups I’, Semigroup Forum 32 (1985), 97–123.CrossRefGoogle Scholar
[17]Rasin, V. V., ‘On the varieties of Cliffordian semigroups’, Semigroups Forum 23 (1981), 201–220.CrossRefGoogle Scholar
[18]Rasin, V. V., ‘Varieties of orthodox Clifford semigroups’, Izv. Vyssh. Uchebn. Zaved. Mat. 26 (1982), 82–85 (in Russian).Google Scholar
[19]Reilly, N. R., ‘Varieties of completely regular semigroups’, J. Austral. Math. Soc. (Series A) 38 (1985), 372–393.CrossRefGoogle Scholar