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Dual symmetric inverse monoids and representation theory

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

D. G. Fitzgerald  and
Jonathan Leech
D. G. Fitzgerald
Affiliation:
School of Mathematics and Physics University of TasmaniaPO BOx 1214 Launceston, Australia7250 e-mail: D.FitzGerald@utas.edu.au
Jonathan Leech
Affiliation:
Department of mathematics Westmont College955 La Paz Road, Santa Barbara California 93108-1099USA e-mail: leech@westmont.edu
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Abstract

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There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid Ix, that is, a monoid of partial one-to-one selfmaps of a set X. The present paper describes the structure of a categorical dual Ix* to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symmetric inverse monoid. It is shown how a representation of S by (full) selfmaps of a set X leads to dual pairs of representations in Ix and Ix*, and how a number of known representations arise as one or the other of these pairs. Conditions on S are described which ensure that representations of S preserve such infima or suprema as exist in the natural order of S. The categorical treatment allows the construction, from standard functors, of representations of S in certain other inverse algebras (that is, inverse monoids in which all finite infima exist). The paper concludes by distinguishing two subclasses of inverse algebras on the basis of their embedding properties.

MSC classification

Secondary: 20M18: Inverse semigroups 20M30: Representation of semigroups; actions of semigroups on sets
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 3 , June 1998 , pp. 345 - 367
Copyright
Copyright © Australian Mathematical Society 1998

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