Inverse 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, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse semigroups and to that of the most recently studied braces, to give a new research perspective to the open problem of finding solutions. Namely, we have recourse to a new structure, the inverse semi-brace, that is a triple (S, +, center dot) with (S, +) a semigroup and (S, center dot) an inverse semigroup satisfying the relation a (b + c) = ab + a (a(-1) + c), for all a, b, c is an element of S, where a(-1) is the inverse of a in (S, center dot). In particular, we give several constructions of inverse semi-braces which allow for obtaining new solutions of the Yang-Baxter equation.