Dual symmetric inverse monoids and
representation theory

D. G. FitzGerald
School of Mathematics and Physics
University of Tasmania
PO Box 1214
Launceston, Australia 7250
e-mail: D.FitzGerald@utas.edu.au

and
Jonathan Leech
Department of Mathematics
Westmont College
955 La Paz Road, Santa Barbara
California 93108-1099 USA
e-mail: leech@westmont.edu

Abstract:

There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid ${\cal I}_X$, that is, a monoid of partial one-to-one selfmaps of a set X. The present paper describes the structure of a categorical dual ${\cal I}_X^{*}$ 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 ${\cal I}_X$ and ${\cal I}_X^{*}$, 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.