Associative Pentagon Algebras

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.