Idempotent set-theoretical solutions of the pentagon equation

TL;DR

This paper investigates idempotent set-theoretical solutions to the pentagon equation, characterizing solutions on monoids with central idempotents and those where a specific map is a monoid homomorphism, revealing structural insights.

Contribution

It provides a comprehensive description of idempotent solutions on monoids with central idempotents and analyzes solutions where a key map is a monoid homomorphism, advancing understanding of the pentagon equation.

Findings

Characterization of idempotent solutions on monoids with central idempotents

Description of solutions where the map θ₁ is a monoid homomorphism

Structural insights into solutions of the pentagon equation

Abstract

A set-theoretical solution of the pentagon equation on a non-empty set $X$ is a function $s:X\times X\to X\times X$ satisfying the relation $s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}$, with $s_{12}=s\times \,id_X$, $s_{23}=id_X \times \, s$ and $s_{13}=(id_X\times \, \tau)s_{12}(id_X\times \,\tau)$, where $\tau:X\times X\to X\times X$ is the flip map given by $\tau(x,y)=(y,x)$, for all $x,y\in X$. Writing a solution as $s(x,y)=(xy ,\theta_x(y))$, where $\theta_x: X \to X$ is a map, for every $x\in X$, one has that $X$ is a semigroup. In this paper, we study idempotent solutions, i.e., $s^2=s$, by showing that the idempotents of $X$ have a key role in such an investigation. In particular, we describe all such solutions on monoids having central idempotents. Moreover, we focus on idempotent solutions defined on monoids for which the map $\theta_1$ is a monoid homomorphism.