Set-theoretical solutions of the pentagon equation on Clifford semigroups

Given a set-theoretical solution of the pentagon equation \(s:S\times S\rightarrow S\times S\) 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 \(\left( S,\,\cdot \right) \) is a semigroup. In this paper, we focus on solutions defined in Clifford semigroups \(\left( S,\,\cdot \right) \) satisfying special properties on the set of all idempotents \({{\,\textrm{E}\,}}(S)\). Into the specific, we provide a complete description of idempotent-invariant solutions, namely, those solutions for which \(\theta _a\) remains invariant in \({{\,\textrm{E}\,}}(S)\), for every \(a\in S\). Moreover, we construct a family of idempotent-fixed solutions, i.e., those solutions for which \(\theta _a\) fixes every element in \({{\,\textrm{E}\,}}(S)\) for every \(a\in S\), from solutions given on each maximal subgroup of S.