Involution semi-braces and the Yang-Baxter equation

Abstract

The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective. We use the new structure of involution semi-brace, that is a quadruple $(S,+,\cdot , ^\ast)$ with $(S,+)$ a semigroup and $(S,\cdot , ^\ast)$ an involution semigroup satisfying the relation $a(b+c)=ab+a(a^\ast+c)$, for all $a,b,c\in S$.