Associative Pentagon Algebras

TL;DR

This paper studies a special class of algebraic structures called associative pentagon algebras, characterizing their properties and providing constructions and examples based on semigroup theory.

Contribution

It introduces and fully characterizes associative pentagon algebras where both operations form semigroups, expanding understanding of solutions to the Pentagon Equation.

Findings

Two families of associative pentagon algebras described

Complete characterization via semigroup equations

Constructed examples and classes provided

Abstract

A set-theoretic solution to the Pentagon Equation can be described as a \emph{pentagon} algebra $(S, \cdot, \ast)$ such that $(S, \cdot)$ is a semigroup and the operations $\cdot$ and $\ast$ are related by two additional equations. This paper aims to investigate associative pentagon algebras in which $(S, \ast)$ is also a semigroup. We introduce and describe two families of associative pentagon algebras which are strongly determined by the properties of the semigroup $(S,\ast)$. We present a complete characterization of such algebras using semigroup equations. We also provide constructions of such associative pentagon algebras and give several classes of examples.