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A combinatorial property of finite full transformation semigroups

Published online by Cambridge University Press:  14 November 2011

John M. Howie
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland, U.K.
Edmund F. Robertson
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland, U.K.
Boris M. Schein
Affiliation:
Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701, U.S.A.

Synopsis

Let E be the set of idempotents in the semigroup Singn of singular self-maps of N = {1, …, n}. Let α ∊ Singn. Then α ∊ E2 if and only if for every x in im α the set −1 either contains x or contains an element of (im α)′.

Write rank α for |im α| and fix α for |{xN: xa = x}|. Define (x, , 2) to be an admissible α-triple if x ∊ (im α)′, xα3xα2. Let comp α (the complexity of α) be the maximum number of disjoint admissible α-triples. Then α ∊ E3 if and only if

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

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