Idempotent set-theoretical solutions of the pentagon equation

A set-theoretical solution of the pentagon equation on a non-empty set $X$ is a function $s:X\times X\to X\times X$ satisfying the relation $s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}$, with $s_{12}=s\times \,id_X$, $s_{23}=id_X \times \, s$ and $s_{13}=(id_X\times \, \tau)s_{12}(id_X\times \,\tau)$, where $\tau:X\times X\to X\times X$ is the flip map given by $\tau(x,y)=(y,x)$, for all $x,y\in X$. Writing a solution as $s(x,y)=(xy ,\theta_x(y))$, where $\theta_x: X \to X$ is a map, for every $x\in X$, one has that $X$ is a semigroup. In this paper, we study idempotent solutions, i.e., $s^2=s$, by showing that the idempotents of $X$ have a key role in such an investigation. In particular, we describe all such solutions on monoids having central idempotents. Moreover, we focus on idempotent solutions defined on monoids for which the map $\theta_1$ is a monoid homomorphism.