Set-theoretical solutions of the pentagon equation on Clifford semigroups

Given a set-theoretical solution of the pentagon equation $s:S\times S\to S\times S$ on a set $S$ and writing $s(a,b)=(a\cdot b,{\theta }_a(b))$, with $\cdot$ a binary operation on $S$ and ${\theta }_a$ a map from $S$ into itself, for every $a\in S$, one naturally obtains that $(S,\cdot )$ is a semigroup. In this paper, we focus on solutions on Clifford semigroups $(S,\cdot )$ satisfying special properties on the set of the idempotents $E(S)$. Into the specific, we provide a complete description of idempotent-invariant solutions, namely, those solutions for which ${\theta }_a$ remains invariant in $E(S)$, for every $a\in S$. Moreover, considering $(S,\cdot )$ as a disjoint union of groups, we construct a family of idempotent-fixed solutions, i.e., those solutions for which ${\theta }_a$ fixes every element in $E(S)$, for every $a\in S$, starting from a solution on each group.