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Nilpotents and congruences on semigroups of transformations with fixed rank

Published online by Cambridge University Press:  14 November 2011

M. Paula O. Marques-Smith  and
R. P. Sullivan
M. Paula O. Marques-Smith
Affiliation:
Departamento de Matematica, Universidade do Minho, 4700 Braga, Portugal
R. P. Sullivan
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia
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Abstract

In 1988, Howie and Marques-Smith studied Pm, a Rees quotient semigroup of transformations associated with a regular cardinal m, and described the elements which can be written as a product of nilpotents in Pm. In 1981, Marques proved that if Δm denotes the Malcev congruence on Pm, then Pm/Δm is congruence-free for any infinite m. In this paper, we describe the products of nilpotents in Pm when m is nonregular, and determine all the congruences on Pm when m is an arbitrary infinite cardinal. We also investigate when a nilpotent is a product of idempotents.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 2 , 1995 , pp. 399 - 412
Copyright
Copyright © Royal Society of Edinburgh 1995

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