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Semimodularity and bisimple ω-semigroups

Published online by Cambridge University Press:  20 January 2009

H. E. Scheiblich
H. E. Scheiblich
Affiliation:
The University of South Carolina
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Let S be a completely 0-simple semigroup and let Λ(S) be the lattice of congruences on S. G. Lallement (2) has described necessary and sufficient conditions on S for Λ(S) to be modular, and has shown that Λ(S) is always semimodular . This result may be stated: If S is 0-bisimple and contains a primitive idempotent, then Λ(S) is semimodular.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 17 , Issue 1 , June 1970 , pp. 79 - 81
Copyright
Copyright © Edinburgh Mathematical Society 1970

References

REFERENCES

(1) Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Vol. 1, Math. Surveys (American Mathematical Society No. 7, Providence, R. I., 1961).Google Scholar
(2) Lallement, G., Demi-groups réguliers, Doctoral Thesis, University of Paris, 1966.Google Scholar
(3) Munn, W. D. and Reelly, N. R., Congruences on a bisimple ω-semigroup, Proc. Glasgow. Math. Assoc. 7 (1966), 184–192.Google Scholar
(4) Munn, W. D., The lattice of congruences on a bisimple ω-semigroup, Proc. Roy. Soc. Edinburgh Sect. A 67 (1965–1967), 175–184.Google Scholar
(5) Reilly, N. R., Bisimple ω-semigroups, Proc. Glasgow Math. Assoc. 7 (1966), 160–169.Google Scholar