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Some subsemigroups of infinite full transformation semigroups

Published online by Cambridge University Press:  14 November 2011

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

As in an earlier paper by the author, three cardinal numbers, the shift, the defect and the collapse, are associated with each element of the full transformation semigroup ℑ(X), where X is an infinite set. It is shown that the elements of finite shift and non-zero defect form a subsemigroup F of ℑ(X). Moreover, if E(F) denotes the set of idempotents in F then 〈E(F)〉 = F, but (E(F))n ⊂F for every finite n. For each infinite cardinal m not exceeding ∣X∣ the set Qm of balanced elements of weight m, i.e. those with shift, defect and collapse all equal to m, forms a subsemigroup of ℑ(X). Moreover, (E(Qm))4=Qm,(E(Qm))3⊂Qm.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 88 , Issue 1-2 , 1981 , pp. 159 - 167
Copyright
Copyright © Royal Society of Edinburgh 1981

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