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A certain sublattice of the lattice of congruences on a regular semigroup

Published online by Cambridge University Press:  24 October 2008

W. D. Munn
W. D. Munn
Affiliation:
University of Glasgow
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Extract

It is well known that, with respect to the natural partial ordering, the set of all congruences on a group forms a modular lattice. In the present paper we develop an extension of this result to the case of a regular semigroup S (α ∈ αSα for all α in S). Let Σ(ℋ) denote the set of all congruences on S with the property that congruent elements generate the same principal left ideal and the same principal right ideal of S. We show (Theorem 1) that, under the natural partial ordering, Σ(ℋ) is a modular lattice with a greatest and a least element.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 3 , July 1964 , pp. 385 - 391
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1) Birkhoff, G. Lattice theory (Revised edition; New York, 1948).Google Scholar
(2) Clifford, A. H. Semigroups admitting relative inverses. Ann. of Math. (2) 42 (1941), 1037–1049.CrossRefGoogle Scholar
(3) Clifford, A. H. and Preston, G. B. The algebraic theory of semigroups, Vol. 1 (American Math. Soc.; Providence, 1961) and Vol. 2 (to appear).Google Scholar
(4) Green, J. A. On the structure of semigroups. Ann. of Math. (2) 54 (1951), 163–172.CrossRefGoogle Scholar
(5) Howie, J. M. The maximum idempotent-separating congruence on an inverse semigroup. Proc. Edinburgh Math. Soc. 14 (1964), 71–79.CrossRefGoogle Scholar
(6) Miller, D. D. and Clifford, A. H. Regular -classes in semigroups. Trans. American Math. Soc. 82 (1956), 270–280.Google Scholar
(7) Munn, W. D. Semigroups satisfying minimal conditions. Proc. Glasgow Math. Assoc. 3 (1957), 145–152.CrossRefGoogle Scholar
(8) Preston, G. B. Congruences on completely 0-simple semigroups. Proc. London Math. Soc. (3) 11 (1961), 557–576.CrossRefGoogle Scholar