Set-theoretic solutions of the Yang–Baxter equation associated to weak braces

We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang–Baxter equation. Specifically, a weak (left) brace is a non-empty set S endowed with two binary operations + and ∘ such that both (S, +) and (S, ∘) are inverse semigroups and a∘(b+c)=(a∘b)-a+(a∘c)anda∘a-=-a+ahold, for all a, b, c∈ S, where - a and a- are the inverses of a with respect to + and ∘ , respectively. In particular, such structures include that of skew braces and form a subclass of inverse semi-braces. Any solution r associated to an arbitrary weak brace S has a behavior close to bijectivity, namely r is a completely regular element in the full transformation semigroup on S× S. In addition, we provide some methods to construct weak braces.