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Products of idempotents in finite full transformation semigroups: some improved bounds

Published online by Cambridge University Press:  14 November 2011

John M. Howie
John M. Howie
Affiliation:
Mathematical Institute, University of St Andrews, St Andrews
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Synopsis

Let E be the set of idempotents in Sn, the semigroup of all singular selfmaps of {1,…, n}. For each α in Sn, there is a unique (κ(α)≧1 such that αψEκ,(α). It is known that κ(α)≦ n + cycl α -fix α, where cyclα is the number of cyclic orbits of a and fix α is the number of fixed points. Equality holds only in the case where a is of rank n – 1. An improved upper bound is obtained for κ(α), applying to elements of arbitrary rank. A lower bound is obtained also.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 98 , Issue 1-2 , 1984 , pp. 25 - 35
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

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