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 x X -> X x X satisfying the relation s(23) s(13) s(12) = s(12) s(23), with s(12) = s x id(X), s(23) = id(X) x s and s(13) = (id(X) x tau)s(12)(id(X) x tau), where tau : X x X -> X x X is the flip map given by tau(x, y) = (y, x), for all x, y is an element of X. Writing a solution as s(x, y) = (xy, theta(x) (y)), where theta(x) : X -> X is a map, for every x is an element of 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 crucial 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.