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Congruences on an orthodox semigroup via the minimum inverse semigroup congruence

Published online by Cambridge University Press:  18 May 2009

Carl Eberhart  and
Wiley Williams
Carl Eberhart
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, U.S.A.
Wiley Williams
Affiliation:
Department of Mathematics, University of Louisville, Louisville, Kentucky 40208, U.S.A.
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It is well known that the lattice Λ(S) of congruences on a regular semigroup S contains certain fundamental congruences. For example there is always a minimum band congruence β, which Spitznagel has used in his study of the lattice of congruences on a band of groups [16]. Of key importance to his investigation is the fact that β separates congruences on a band of groups in the sense that two congruences are the same if they have the same meet and join with β. This result enabled him to characterize θ-modular bands of groups as precisely those bands of groups for which ρ⃗(ρ∨β, ρ∧β)is an embedding of Λ(S) into a product of sublattices.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 18 , Issue 2 , July 1977 , pp. 181 - 192
Copyright
Copyright © Glasgow Mathematical Journal Trust 1977

References

REFERENCES

Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vols. 1 and 2, Amer. Math. Soc. Mathematical Surveys No. 7 (Providence, R.I., 1961, 1967).Google Scholar
Green, D. G., The lattice of congruences on an inverse semigroup, Pacific J. Math. 57 (1975), 141–152.CrossRefGoogle Scholar
Hall, T. E., On regular semigroups whose idempotents form a subsemigroup. Bull. Austral. Math. Soc. 1 (1969), 195–208.CrossRefGoogle Scholar
Hall, T. E., On the lattice of congruences on a regular semigroup, Bull. Austral. Math. Soc. 1 (1969), 231–235.CrossRefGoogle Scholar
Howie, J. M., Naturally ordered bands, Glasgow Math. J. 8 (1967), 55–58.CrossRefGoogle Scholar
Howie, J. M., The maximum idempotent-separating congruence on an inverse semigroup, Proc. Edinburgh Math. Soc. (2) 14 (1964–5), 71–79.CrossRefGoogle Scholar
Howie, J. M. and Lallement, G., Certain fundamental congruences on a regular semigroup, Proc. Glasgow Math. Assoc. 7 (1966), 145–159.CrossRefGoogle Scholar
Kapp, K. M. and Schneider, H., Completely 0-simple semigroups (New York, 1969).Google Scholar
Lallement, G., Congruences et equivalences de Green sur un demi-groupe regulier, C. R. Acad. Sci. Sér A, 262 (1966), 613–616.Google Scholar
Meakin, J., Congruences on orthodox semigroups, J. Austral. Math. Soc. 12 (1971), 323–341.Google Scholar
Munn, W. D., A certain sublattice of the lattice of congruences on a regular semigroup, Proc. Cambridge Philos. Soc. 60 (1964), 385–391.Google Scholar
Preston, G. B., Congruences on completely simple semigroups, Proc. London Math. Soc. (3) 11 (1961), 557–576.CrossRefGoogle Scholar
Reilly, N. R. and Scheiblich, H. E., Congruences on regular semigroups, Pacific J. Math. 23 (1967), 349–360.CrossRefGoogle Scholar
Scheiblich, H. E., Certain congruence and quotient lattices related to completely 0-simple and primitive regular semigroups, Glasgow Math. J. 10 (1969), 21–24.Google Scholar
Scheiblich, H. E., Kernels of inverse semigroup homomorphisms, J. Austral. Math. Soc. 18 (1974), 289–292.Google Scholar
Spitznagel, C., The lattice of congruences on a band of groups, Glasgow Math. J. 14 (1973), 187–197.CrossRefGoogle Scholar
Spitznagel, C., θ-modular bands of groups, Trans. Amer. Math. Soc. 177 (1973), 469–482.Google Scholar
Fergenbaum, Ruth, Kernels of orthodox semigroup homomorphisms, J. Austral. Math. Soc. Ser. A 22 (1976), 234–245.Google Scholar