Dual symmetric inverse monoids and representation theory

Abstract

There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid ^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 J$ 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 J x and J^, 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.1991 Mathematics subject classification (Amer. Math. Soc): primary 20M18; secondary 2OM3O.

Generation of infinite factorizable inverse monoids

“…The factorizable part F(I X ) of I X is therefore equal to the set of all restrictions of all permutations of X to subsets of X. As in [13], we write F X = F(I X ).…”

Factorizable inverse monoids

“…(The proof of this is straightforward; one need only remember that ef = e _ f; the intersection of all equivalences containing both e and f .) The resulting factorizable inverse monoid is the factorizable part F (I X ) of the dual symmetric inverse monoid ( [10], Prop.…”

“…eqn. (4.2)) and so is a submonoid of F I jV j : Moreover, for …nite-dimensional V; it can be identi…ed as the dual partial automorphism monoid of V ; since the category of linear spaces and maps is self-dual, S is isomorphic with PA (V ) : The general set-up is described passim in sections 1 and 5 of [10], but it may be of interest to exhibit an explicit isomorphism. For this, …rst equip V with an inner product h_ j _i : Then note that (W + Z) ?…”

On presentations of Brauer-type monoids

“…, λ k ), of positive integers such that λ 1 ≥ λ 2 ≥ · · · ≥ λ k and λ 1 + · · · + λ k = n). The types of the elements classify the D-classes in IT n , see [FL,Section 3].…”