Set-theoretical solutions of the pentagon equation on Clifford semigroups

Abstract

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

On the Structure of Set-Theoretic Polygon Equations

“…for all a, b, c ∈ U. Set-theoretic solutions have been obtained in [1,5,6,7,22,25,26,27,28,31,32,45,46,70].…”