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

Abstract

We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang-Baxter equation. Specifically, a skew (left) inverse semi-brace is a non-empty set S endowed with two binary operations + and • such that both (S, +) and (S, •) are inverse semigroups and they holdfor 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 skew inverse semi-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 skew inverse semi-braces.