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Idempotent rank in finite full transformation semigroups

  • John M. Howie (a1) and Robert B. McFadden (a2)

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.

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1Cohen, Daniel I. A.. Elementary Principles of Combinatorial Theory (New York: Wiley, 1978).
2Evseev, A. E. and Podran, N. E.. Semigroups of transformations generated byidempotents of given defect. Izv. Vyssh. Uchebn. Zaved. Mat. 2 (117)(1972), 4450.
3Gomes, G. M. S. and Howie, J. M.. On the rank of certain semigroups of transformations. Math. Proc. Cambridge Philos. Soc. 101 (1987), 395403.
4Howie, John M.. An introduction to semigroup theory (New York: Academic Press, 1976).

Idempotent rank in finite full transformation semigroups

  • John M. Howie (a1) and Robert B. McFadden (a2)

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