Set-theoretical solutions of the pentagon equation on Clifford semigroups

Given a set-theoretical solution of the pentagon equation s : S × S → S × S on a set S and writing s(a, b) = (a · b, θa(b)), with · a binary operation on S and θa a map from S into itself, for every a ∈ S, one naturally obtains that (S, ·) is a semigroup. In this paper, we focus on solutions defined in Clifford semigroups (S, ·) satisfying special properties on the set of all idempotents E(S). Into the specific, we provide a complete description of idempotent-invariant solutions, namely, those solutions for which θa remains invariant in E(S), for every a ∈ S. Moreover, we construct a family of idempotent-fixed solutions, i.e., those solutions for which θa fixes every element in E(S) for every a ∈ S, from solutions given on each maximal subgroup of S.