Des Fitzgerald: Representing inverse semigroups in complete inverse algebras

The Wagner-Preston theorem says that any inverse semigroup may be embedded in the so-called symmetric inverse semigroup consisting of all injective partial maps between subsets of some set.  This is a very good representation theorem: it illuminates, for example, all the order properties important in the study of inverse semigroups.  On the other hand, the embedding of the proof is neither unique nor necessarily optimal.  So it is desirable to be able to describe and classify all representations of inverse semigroups in a symmetric one.  A theorem of Schein does exactly that, and work of Munn and Ponizovskii does likewise for linear (or matrix) representations of inverse semigroups.

Beyond the symmetric inverse semigroups, there are many concrete inverse semigroups (made up of partial automorphisms of various structures, for example) which may serve as useful models of inverse semigroups.  Yet our means of constructing good representations of this nature is embarrassingly limited: we are mostly reliant on Cayley’s theorem in semigroup theory, and its modifications, to make representations by injective partial maps, matrices and the like.  (This may be contrasted with the situation in group theory, where notable groups have been described as automorphism groups of geometries, graphs, spaces, etc.)

This talk will illustrate some of these remarks, and set out a partial description and classification of representations of arbitrary inverse semigroups in members of a fairly broad class of (what can be thought of as) partial automorphism monoids.  Thus it may be seen as a generalisation of Schein’s approach.  We have to consider the meaning of transitivity and effectiveness in this more general setting, and we pay some particular attention to the Boolean inverse monoids and the dual symmetric inverse monoids as receivers for representations.