Involution semi-braces and the Yang-Baxter equation

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,+,.,*) with (S,+) a semigroup and (S,.,*) an involution semigroup satisfying the relation a(b+c)=ab+a(a*+c), for all a,b,cϵS.