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

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 $\mathbb{C}P{T}_n$, the algebra of the monoid of all partial functions on $n$ elements. Our description uses an isomorphism between $\mathbb{C}P{T}_n$ and the algebra of the epimorphism category, $E_n$, whose objects are the subsets of $\{1,\dots ,n\}$ and morphism are all total epimorphisms. This is an extension of a well known isomorphism of the algebra of $I{S}_n$ (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 $\mathbb{C}P{T}_n$ has three blocks for $n>1$ and we give a natural description of the descending Loewy series of $\mathbb{C}P{T}_n$ in the category form.