Is there a nice theory of representations of inverse semigroups in special inverse algebras? a discussion paper

Abstract

I consider the problem of elaborating an analogue, for the dual symmetric inverse monoid, of the `classical' (Schein) theory of representations in the (primal) symmetric inverse monoid. This requires appropriate analogues of the concepts of 'effective' and 'transitive' partial bijections for block multipermutations. I use examples to show that it is not at first clear what these should be, and argue that an adequate solution of this issue would also be a useful test case in developing theories for representation in partial automorphism monoids of entities such as (perhaps) graphs, modules, etc.