  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
$(S,+, \cdot)$ with $(S,+)$ a semigroup and $(S, \cdot)$ an inverse semigroup
satisfying the relation $a \left(b + c\right) = a b + a\left(a^{-1} +
c\right)$, for all $a,b,c \in S$, where $a^{-1}$ is the inverse of $a$ in $(S,
\cdot)$. In particular, we give several constructions of inverse semi-braces
which allow for obtaining solutions that are different from those until known.
