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

A dual weak brace is an algebraic structure $(S,+,\circ )$ 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., ${\mathcal{D}}_r(S)=\{z\in S\mid \mathrm{\forall }a,b\in S(a+b)\circ z=a\circ z-z+b\circ z\}$, that is a full inverse subsemigroup of $(S,\circ )$. Regarding $S$ as a strong semilattice $[Y,B_{\alpha },{\phi }_{\alpha ,\beta }]$ of skew braces $B_{\alpha }$, we analyze when ${\mathcal{D}}_r(S)=\underset{\alpha \in Y}{}{\mathcal{D}}_r(B_{\alpha })$ and in which cases a deformed solution is the strong semilattices of deformed solutions.