Let $\mathcal{C}_n$ be the Catalan monoid of self-maps, that is, $\alpha$ of $\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$ and decreasing ($\forall x$, $\alpha(x)\le x$). For any semigroup $S$, all idempotent elements in $S$ is denoted by $E(S)$.
For any idempotent element of $\mathcal{C}_n$, say $\epsilon$, let $\mathcal{C}_n(\epsilon)=\{\alpha\in\mathcal{C}_n: \alpha^m=\epsilon, m\in \mathbb{Z}^+\}$. Let $T=\cup_{\epsilon\in E(T)}\mathcal{C}_n(\epsilon)$.
- My question is when $T$ can be subsemigroup of $\mathcal{C}_n$?
It is clear that if $E(T)$ consist only one element, then $T$ is a subsemigroup of $\mathcal{C}_n$. But i need more general examination.