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 s_23 s_13 s_12 = s_12 s_13 s_23, where s_12=s× id _S , s_23= id _S× s , and s_13=( id _S×τ ) s_12 ( id _S×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 s_23 s_13 s_12=s_12 s_23. 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.