# Des Fitzgerald: Representing inverse semigroups in complete inverse algebras

Datum objave: 14. 5. 2018

Vir: Seminar za algebro in funkcionalno analizo

Vir: Seminar za algebro in funkcionalno analizo

Četrtek, 17. 5. 2018, ob 12:30 v predavalnici 2.04, FMF, Jadranska 21, Ljubljana

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.