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The lattice of congruences on a band of groups

Published online by Cambridge University Press:  18 May 2009

C. Spitznagel
C. Spitznagel
Affiliation:
University of Kentucky, Lexington, Kentucky 40506, and John Carroll University, Cleveland, Ohio 44118
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It is implicit in a result of Kapp and Schneider [3] that, if Sisa completely simple semigroup, then the lattice Λ(S) of congruences on S can be embedded in the product of certain sublattices. In this paper we consider the problem of embedding Λ(S) in a product of sublattices, when S is an arbitrary band of groups. The principal tool is the θ-relation of Reilly and Scheiblich [7]. The class of θ-modular bands of groups is definedby means of a type of modularity condition on Λ(S). It is shown that the θ-modular bands of groups are precisely those for which a certain function is an embedding of Λ(S) into a product of sublattices. The problem of embedding the inverse semigroup congruences into a certain product lattice is also considered.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 14 , Issue 2 , September 1973 , pp. 187 - 197
Copyright
Copyright © Glasgow Mathematical Journal Trust 1973

References

REFERENCES

1.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. I, Amer. Math. Soc. Mathematical Surveys No.7 (Providence, R.I., 1961).Google Scholar
2.Howie, J. M. and Lallement, G., Certain fundamental congruences on a regular semigroup, Proc. Glasgow Math. Assoc. 7 (1966), 145–159.CrossRefGoogle Scholar
3.Kapp, K. M. and Schneider, H., Completely O-simple semigroups(New York, 1969).Google Scholar
4.Lallement, G., Congruence et equivalences de Green sur un demi-group regulier, C. R. Acad. Sci. Paris, Serie A, 262 (1966), 613–616.Google Scholar
5.Leech, J., The structure of bands of groups; to appear.Google Scholar
6.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
7.Reilly, N. R. and Scheiblich, H. E., Congruences on regular semigroups, Pacific J. Math. 23 (1967), 349–360.CrossRefGoogle Scholar
8.Scheiblich, H. E., Certain congruence and quotient lattices related to completely 0-simple and primitive regular semigroups, Glasgow Math. J. 10 (1969), 21–24.CrossRefGoogle Scholar
9.Spitznagel, C., θ-modular bands of groups, Trans. Amer. Math. Soc; to appear.Google Scholar