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

TL;DR

This paper describes the quiver structure of the algebra of all partial functions on a set, using category theory and EI-category algebra techniques, and explores its block structure and Loewy series.

Contribution

It extends known isomorphisms to describe the quiver of the algebra of all partial functions and related monoids as EI-category algebras, providing new structural insights.

Findings

The algebra $\mathbb{C}PT_{n}$ has three blocks for $n>1$.

The quiver of $\mathbb{C}PT_{n}$ is explicitly described using EI-category techniques.

The Loewy series of $\mathbb{C}PT_{n}$ is characterized in category form.

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