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 weak (left) brace is a non-empty set $S$ endowed with two binary operations $+$ and $\circ$ such that both $(S,+)$ and $(S, \circ)$ are inverse semigroups and they hold \begin{align*} a \circ \left(b+c\right) = a\circ b - a +a\circ c \qquad \text{and} \qquad a\circ a^- = - a + a, \end{align*} for all $a,b,c \in S$, where $-a$ and $a^-$ are the inverses of $a$ with respect to $+$ and $\circ$, 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 $S\times S$. In addition, we provide some methods to construct weak braces.