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On a sublattice of the lattice of congruences on a simple regular ω-semigroup

Published online by Cambridge University Press:  09 April 2009

G. R. Baird
G. R. Baird
Affiliation:
Department of Mathematics, Monash University
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The set E of idemponents of a semigroup S can be partially ordered by defining e ≦f and only if ef = fe = e (e,f ∈ E). If E = {ei: i = i = 0,1…} and under this ordering e0 > e1 > e2… then we call S an ω-semigroup. Munn [10] has given a complete classification of simple regular ω-semigroups in terms of groups and group homomorphisms. Let ∧0(S) denote the set of congruences on a simple regular w-semigroup S consisting of those congruences which either are idempotent-separating or are group congruences on S. It is evident that ∧0(S) is a sublattice of the lattice of all congruences on S.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 4 , June 1972 , pp. 461 - 471
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Volume I (Math. Surveys, Number 7, Amer. Math. Soc. 1961).Google Scholar
[2]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Volume II (Math. Surveys, Number 7, Amer. Math. Soc. 1967).Google Scholar
[3]Hall, M. Jnr, The theory of Groups (The Macmillan Company, New York, 1959).Google Scholar
[4]Howie, J. M., ‘The maximum idempotent-separating congruence on an inverse semigroup’, Proc. Edinburgh Math. Soc. 14 (1964), 71–79.Google Scholar
[5]Lallement, G., ‘Congruences et équivalences de Green sur un demi-groupe régulier’, C. R. Acad. Sc. Paris Série A262 (1966), 613–616.Google Scholar
[6]Munn, W. D., ‘A class of irreducible matrix representations of an arbitrary inverse semigroup’, Proc. Glasgow Math. Assoc. 5 (1961), 41–48.CrossRefGoogle Scholar
[7]Munn, W. D., ‘A certain sublattice of the congruences on a regular semigroup’, Proc. Camb. Phil. Soc. 60 (1964), 385–391.Google Scholar
[8]Munn, W. D. and Reilly, N. R., ‘Congruences on a bisimple ω-semigroup’, Proc. Glasgow Math. Assoc. 7 (1966), 184–192.Google Scholar
[9]Munn, W. D., ‘The lattice of congruences on a bisimple ω-semigroup’, Proc. Roy. Soc. Edinburgh 67 (1966), 175–184.Google Scholar
[10]Munn, W. D., ‘Regular ω-semigroups’, Glasgow Math. J. 9 (1968), 46–66.CrossRefGoogle Scholar
[11]Reilly, N. R., ‘Bisimple ω-semigroups’, Proc. Glasgow Math. Assoc. 7 (1966), 160–167.Google Scholar