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Certain fundamental congruences on a regular semigroup†

Published online by Cambridge University Press:  18 May 2009

J. M. Howie  and
G. Lallement
J. M. Howie
Affiliation:
University of GlasgowGlasgow, W. 2
G. Lallement
Affiliation:
Institut Henri PoincaréParis
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In recent developments in the algebraic theory of semigroups attention has been focussing increasingly on the study of congruences, in particular on lattice-theoretic properties of the lattice of congruences. In most cases it has been found advantageous to impose some restriction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating congruence on a regular or inverse semigroup by the authors separately [9, 10] and by Munn [14], and the minimum semilattice congruence on a general or commutative semigroup by Tamura and Kimura [19], Yamada [22], Clifford [3] and Petrich [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall consistently denote by α β and η respectively.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 7 , Issue 3 , January 1966 , pp. 145 - 159
Copyright
Copyright © Glasgow Mathematical Journal Trust 1966

References

REFERENCES

1.Clifford, A. H., Semigroups admitting relative inverses, Ann. of Math. 42 (1941), 1037–1049.CrossRefGoogle Scholar
2.Clifford, A. H., Bands of semigroups, Proc. Amer. Math. Soc. 5 (1954), 499–504.CrossRefGoogle Scholar
3.Clifford, A. H., Review of Yamada's paper [22], Math. Rev. 17 (1956), 584.Google Scholar
4.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, American Mathematical Society Mathematical Surveys No. 7, Vol. 1 (Providence, R. I., 1961).Google Scholar
5.Dubreil, P., Contribution à la théorie des demi-groupes, Mém. Acad. Sci. Inst. France (2) 63 (1941), no. 3, 1–52.Google Scholar
6.Fantham, P. H. H., On the classification of a certain type of semigroup, Proc. London Math. Soc. (3) 10 (1960), 409–427.CrossRefGoogle Scholar
7.Green, J. A., On the structure of semigroups, Ann, of Math. 54 (1951), 163–172.CrossRefGoogle Scholar
8.Green, J. A. and Rees, D., On semigroups in which x r = x, Proc. Cambridge Philos. Soc. 48 (1952), 35–40.CrossRefGoogle Scholar
9.Howie, J. M., The maximum idempotent-separating congruence on an inverse semigroup Proc. Edinburgh Math. Soc. 14 (1964), 71–79.CrossRefGoogle Scholar
10.Lallement, G., Congruences et équivalences de Green sur un demi-groupe régulier (to appear).Google Scholar
11.McLean, D., Idempotent semigroups, Amer. Math. Monthly 61 (1954), 110–113.CrossRefGoogle Scholar
12.Miller, D. D. and Clifford, A. H., Regular -classes in semigroups, Trans. Amer. Math. Soc. 82 (1956), 270–280.Google Scholar
13.Munn, W. D., A class of irreducible matrix representations of an arbitrary inverse semigroup, Proc. Glasgow Math. Assoc. 5 (1961), 41–48.CrossRefGoogle Scholar
14.Munn, W. D., A certain sublattice of the lattice of congruences on a regular semigroup, Proc. Cambridge Philos. Soc. 60 (1964), 385–391.CrossRefGoogle Scholar
15.Petrich, Mario, The maximal semilattice decomposition of a semigroup, Math. Zeit. 85 (1964), 68–82.CrossRefGoogle Scholar
16.Preston, G. B., Inverse semi-groups, J. London Math. Soc. 29 (1954), 396–403.CrossRefGoogle Scholar
17.Preston, G. B., Congruences on Brandt semigroups, Math. Ann. 139 (1959), 91–94.CrossRefGoogle Scholar
18.Rees, D., On semi-groups, Proc. Cambridge Philos. Soc. 36 (1940), 387–400.CrossRefGoogle Scholar
19.Tamura, T. and Kimura, N., On decompositions of a commutative semigroup, Kodai Math. Sem. Rep. 4 (1954), 109–112.Google Scholar
20.Teissier, Marianne, Sur les équivalences régulières dans les demi-groupes, C. R. Acad. Sci. Paris 232 (1951), 1987–1989.Google Scholar
21.Vagner, V. V., The theory of generalised groups and generalised heaps, Mat. Sborn. (N. S.) 32 (1953), 545–632 (Russian).Google Scholar
22.Yamada, M., On the greatest semilattice decomposition of a semigroup, Kodai Math. Sem. Rep. 7 (1955), 59–62.CrossRefGoogle Scholar