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Idempotent rank in finite full transformation semigroups

Published online by Cambridge University Press:  14 November 2011

John M. Howie  and
Robert B. McFadden
John M. Howie
Affiliation:
Department of Mathematical Sciences, University of St Andrews, St Andrews, Scotland, KY16 9SS, U.K
Robert B. McFadden
Affiliation:
Department of Mathematics, University of Louisville, Louisville, Ky 40292, U.S.A.
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Synopsis

The subsemigroup Singn of singular elements of the full transformation semigroup on a finite set is generated by n(n − l)/2 idempotents of defect one. In this paper we extend this result to the subsemigroup K(n, r) consisting of all elements of rank r or less. We prove that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n, r), the Stirling number of the second kind.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 3-4 , 1990 , pp. 161 - 167
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Cohen, Daniel I. A.. Elementary Principles of Combinatorial Theory (New York: Wiley, 1978).Google Scholar
2Evseev, A. E. and Podran, N. E.. Semigroups of transformations generated byidempotents of given defect. Izv. Vyssh. Uchebn. Zaved. Mat. 2 (117)(1972), 44–50.Google Scholar
3Gomes, G. M. S. and Howie, J. M.. On the rank of certain semigroups of transformations. Math. Proc. Cambridge Philos. Soc. 101 (1987), 395–403.CrossRefGoogle Scholar
4Howie, John M.. An introduction to semigroup theory (New York: Academic Press, 1976).Google Scholar