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

We investigate a new algebraic structure which always gives rise to set-theoretic solution of the Yang-Baxter equation. Specifically, weak (left) brace is non-empty set $S$ endowed with two binary operations $+$ and $\circ$ such that both $(S,+)$ $(S, \circ)$ are inverse semigroups they hold \begin{align*} \circ \left(b+c\right) = a\circ b - +a\circ c \qquad \text{and} a^- + a, \end{align*} for all $a,b,c \in S$, where $-a$ $a^-$ inverses $a$ respect $\circ$, respectively. In particular, structures include skew braces form subclass semi-braces. Any $r$ associated an arbitrary has behavior close bijectivity, namely completely regular element in full transformation semigroup on $S\times S$. addition, we provide some methods construct braces.