Dual Symmetric Inverse Monoids and Representation Theory
نویسندگان
چکیده
There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoidIX , that is, a monoid of partial one-to-one selfmaps of a set X . The present paper describes the structure of a categorical dual I Ł X 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 IX and I Ł X , 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 20M30. 1. Background information In this paper we consider (i) the dual symmetric inverse monoid I Ł X of all bijections between the quotient sets of a given set X , and more generally, the dual symmetric inverse monoid I Ł A of all isomorphisms between the quotient objects of an object A in any sufficiently well-endowed category; and (ii) representations of arbitrary inverse monoids in dual symmetric inverse monoids. To do so with sufficient generality requires that we first recall the category-theoretic framework of symmetric inverse monoids. This approach directs consideration to both duality, and to the existence of extra structure (that of complete inverse algebras) in both the symmetric and dual symmetric cases. c 1998 Australian Mathematical Society 0263-6115/98 $A2.00 + 0.00
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