Deformed solutions of the Yang-Baxter equation associated to dual weak braces

Abstract

A dual weak brace is an algebraic structure $\left(S,\,+,\,\circ\right)$ including skew braces and giving rise to a set-theoretic solution of the Yang-Baxter equation. We show that such a map belongs to a family of set-theoretic solutions, called deformed solutions, that are defined on $S$ and depending on certain parameters. We prove these elements are exactly those belonging to the distributor of $S$, i.e., $\mathcal{D}_r(S)=\{z \in S \, \mid \, \forall \, a,b \in S \quad (a+b) \circ z=a\circ z-z+b \circ z\}$, that is a full inverse subsemigroup of $\left(S, \circ\right)$. Regarding $S$ as a strong semilattice $[Y, B_\alpha, \phi_{\alpha,\beta}]$ of skew braces $B_\alpha$, we analyze when $\mathcal{D}_r(S)=\mathop{\dot{\bigcup}}\limits_{\alpha\in Y} \mathcal{D}_r(B_\alpha)$ and in which cases a deformed solution is the strong semilattices of deformed solutions.