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

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 $f(x)\le f(y)$ if $x\le y$) the monoid of all order-decreasing functions (i.e. functions satisfying $f(x)\le x$) 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.