  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.
