Let $C_n$ be the monoid consisting of monotone maps $\{1,...,n\} \rightarrow \{1,...,n\}$ with $f(i) \leq i$ for all $i$. The cardinality of $C_n$ is given by the Catalan numbers.

Consider $A_n= \mathbb{C} C_n$ , the monoid algebra over the complex numbers.

Some experiments suggest that $A_n$ is (isomorphic to) a quiver algebra $KQ/I$ with admissible ideal $I$ having $2^{n-1}$ simple modules. (It seems $A_n/rad(A_n) \cong \mathbb{C}^{2^{n-1}}$, which would imply this. This is true for $n \leq 5$.)

Question: Is such an explicit description via quiver and admissible relations known for this monoid algebra?

Answers for small $n$ would also be interesting. For $n=2$ the algebra is isomorphic to $\mathbb{C} \times \mathbb{C}$.