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

Inverse semigroups generalise both groups and semilattices, and describe partial symmetries just as groups do for total symmetries; they also arise naturally in appropriately rich categories, essentially as monoids of partial automorphisms. Analogy with the group case suggests that the use of inverse semigroups will be assisted by the development of a workable representation theory. Moreover the representation theory of some operator algebras turns out to be linked with that of inverse semigroups [refs]. We do have one situation where, thanks to Boris Schein’s work, the theory is well-developed: e¤ective representations in the symmetric inverse monoid IX decompose to a ‘sum’of transitive ones, and every transitive one has an ‘internal’description in terms of appropriately de…ned cosets of closed inverse subsemigroups [ref]. In [3] the authors introduced the categorical dual of IX ; named as ‘the’dual symmetric inverse monoid, I X ; but also describable as the inverse semigroup of all block multipermutations of a set X; and as a semigroup of special binary relations with a variant multiplication. Maltcev [4] has independently discovered this semigroup as, apart from an extra zero, a maximal inverse subsemigroup of the composition (or partition) monoid, and so calls it the inverse partition monoid. Surely the protean character of this monoid argues for its importance. Maltcev also describes maximal