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On the ranks of certain finite semigroups of transformations

Published online by Cambridge University Press:  24 October 2008

Gracinda M. S. Gomes  and
John M. Howie
Gracinda M. S. Gomes
Affiliation:
Faculdade de Ciencias, Universidade de Lisboa
John M. Howie
Affiliation:
Mathematical Institute, University of St Andrews
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Extract

It is well-known (see [2]) that the finite symmetric group Sn has rank 2. Specifically, it is known that the cyclic permutations

generate Sn,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n (< ∞) symbols has rank 3, being generated by the two generators of Sn, together with an arbitrarily chosen element of defect 1. (See Clifford and Preston [1], example 1.1.10.) The rank of Singn, the semigroup of all singular self-maps of {1, …, n}, is harder to determine: in Section 2 it is shown to be ½n(n − 1) (for n ≽ 3). The semigroup Singn it is known to be generated by idempotents [4] and so it is possible to define the idempotent rank of Singn as the cardinality of the smallest possible set P of idempotents for which <F> = Singn. This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 3 , May 1987 , pp. 395 - 403
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

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