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

中文翻译：

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局部平凡范畴代数的部分函数的与顺序有关的对分形的表示理论

在本文中，我们研究了n个集合上的三个部分函数的半群象形的表示理论。的所有的保持顺序的功能的半群（即，满足功能˚F（X）≤ ˚F（Ý）如果X ≤ ÿ）的所有顺序递减函数的半群（即满足功能˚F（X）≤ X）和它们的交集（也称为加泰罗尼亚语部分半体）。我们在这些等分体的代数和某些相应的局部琐碎类别的代数之间使用同构。我们获得每个代数颤动表示的明确描述。此外，我们描述了其他不变式，例如Cartan矩阵和Loewy长度。