# Representation theory of order-related monoids of partial functions as locally trivial category algebras

## TL;DR

This paper explores the representation theory of three monoids of partial functions on an n-set, revealing their algebraic structures through locally trivial category algebras and providing explicit descriptions of their quivers and invariants.

## Contribution

It introduces an isomorphism between monoid algebras and locally trivial category algebras, offering explicit quiver presentations and invariants for these structures.

## Key findings

- Explicit quiver presentations for each algebra
- Descriptions of Cartan matrices and Loewy lengths
- Isomorphism between monoid algebras and category algebras

## Abstract

In this paper we study the representation theory of three monoids of partial functions on an $n$-set. The monoid of all order-preserving functions (i.e., functions satisfying $f(x)\leq f(y)$ if $x\leq y$) the monoid of all order-decreasing functions (i.e. functions satisfying $f(x)\leq x$) and their intersection (also known as the partial Catalan monoid). We use an isomorphism between the algebras of these monoids and the algebras of some corresponding locally trivial categories. We obtain an explicit description of a quiver presentation for each algebra. Moreover, we describe other invariants such as the Cartan matrix and the Loewy length.