The representation theory of the monoid of all partial functions on a set and related monoids as EI-category algebras

Abstract

The (ordinary) quiver of an algebra A is a graph that contains information about the algebra's representations. We give a description of the quiver of CPTn, the algebra of the monoid of all partial functions on n elements. Our description uses an isomorphism between CPTn and the algebra of the epimorphism category, En, whose objects are the subsets of {1,…,n} and morphism are all total epimorphisms. This is an extension of a well known isomorphism of the algebra of ISn (the monoid of all partial injective maps on n elements) and the algebra of the groupoid of all bijections between subsets of an n-element set. The quiver of the category algebra is described using results of Margolis, Steinberg and Li on the quiver of EI-categories. We use the same technique to compute the quiver of other natural transformation monoids. We also show that the algebra CPTn has three blocks for n>1 and we give a natural description of the descending Loewy series of CPTn in the category form.