Inverse semi-braces and the Yang-Baxter equation

The main aim of this paper is to provide set-theoretical solutions the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In specific, we draw on both classical theory inverse semigroups and most recently studied braces, give a research perspective open problem finding solutions. Namely, have recourse structure, semi-brace, triple (S,+,?) with (S,+) semigroup (S,?) an satisfying relation a(b+c)=ab+a(a?1+c), for all a,b,c?S, where a?1 in (S,?). particular, several constructions semi-braces which allow obtaining equation.