Set-theoretical solutions of the pentagon equation on Clifford semigroups

TL;DR

This paper classifies special set-theoretical solutions to the pentagon equation on Clifford semigroups, focusing on idempotent-invariant and idempotent-fixed solutions, and constructs solutions based on group decompositions.

Contribution

It provides a complete description of idempotent-invariant solutions and constructs idempotent-fixed solutions on Clifford semigroups, advancing understanding of the pentagon equation in this context.

Findings

Classified idempotent-invariant solutions on Clifford semigroups.

Constructed idempotent-fixed solutions from solutions on groups.

Connected solutions to the structure of Clifford semigroups.

Abstract

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 $\left(S,\,\cdot\right)$ is a semigroup. In this paper, we focus on solutions on Clifford semigroups $\left(S,\,\cdot\right)$ 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.