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A nilpotent-generated semigroup associated with a semigroup of full transformations

Published online by Cambridge University Press:  14 November 2011

John M. Howie  and
M. Paula O. Marques-Smith
John M. Howie
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews KY16 9SS, Scotland, U.K.
M. Paula O. Marques-Smith
Affiliation:
Area de Matematica, Universidade do Minho, 4700 Braga, Portugal
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Synopsis

Let X be a set with infinite regular cardinality m and let ℱ(X) be the semigroup of all self-maps of X. The semigroup Qm of ‘balanced’ elements of ℱ(X) plays an important role in the study by Howie [3,5,6] of idempotent-generated subsemigroups of ℱ(X), as does the subset Sm of ‘stable’ elements, which is a subsemigroup of Qm if and only if m is a regular cardinal. The principal factor Pm of Qm, corresponding to the maximum ℱ-class Jm, contains Sm and has been shown in [7] to have a number of interesting properties.

Let N2 be the set of all nilpotent elements of index 2 in Pm. Then the subsemigroup (N2) of Pm generated by N2 consists exactly of the elements in Pm/Sm. Moreover Pm/Sm has 2-nilpotent-depth 3, in the sense that

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 1-2 , 1988 , pp. 181 - 187
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

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