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On inverses of products of idempotents in regular semigroups

Published online by Cambridge University Press:  09 April 2009

D. G. Fitz-Gerald
D. G. Fitz-Gerald
Affiliation:
Department of Mathematics Monash University, Clayton
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Let E be the set of idempotents of a regular semigroup; we prove that V(En) = En+1 (see below for the meaning of this notation). This generalizes a result of Miller and Clifford ([3], theorem 4, quoted as exercise 3(b), p. 61, of Clifford and Preston [1]) and the converse, proved by Howie and Lallement ([2], lemma 1.1), which together establish the case n = 1. As a corollary, we deduce that the subsemigroup generated by the idempotents of a regular semigroup is itself regular.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 3 , May 1972 , pp. 335 - 337
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Volume 1, (Math. Surveys, No. 7, Amer. Math. Soc., 1961).CrossRefGoogle 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]Miller, D. D. and Clifford, A. H., ‘Regular D-classes in semigroups’, Trans. Amer. Math. Soc. 82 (1956), 270–280.Google Scholar