Set-theoretical solutions of the Yang–Baxter and pentagon equations on semigroups

The Yang–Baxter and pentagon equations are two well-known equations of Mathematical Physic. If S is a set, a map s: S× S→ S× S is said to be a set-theoretical solution of the quantum Yang–Baxter equation if s23s13s12=s12s13s23,where s12=s×idS, s23=idS×s, and s13=(idS×τ)s12(idS×τ) and τ is the flip map, i.e., the map on S× S given by τ(x, y) = (y, x). Instead, s is called a set-theoretical solution of the pentagon equation if s23s13s12=s12s23.The main aim of this work is to display how solutions of the pentagon equation turn out to be a useful tool to obtain new solutions of the Yang–Baxter equation. Specifically, we present a new construction of solutions of the Yang–Baxter equation involving two specific solutions of the pentagon equation. To this end, we provide a method to obtain solutions of the pentagon equation on the matched product of two semigroups, that is a semigroup including the classical Zappa product.