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

TL;DR

This paper explores deformed solutions of the Yang-Baxter equation derived from dual weak braces, characterizing their structure and conditions under which they form strong semilattices of solutions.

Contribution

It introduces the concept of deformed solutions associated with dual weak braces and analyzes their algebraic structure and conditions for forming strong semilattices.

Findings

Deformed solutions belong to the distributor of the dual weak brace.

The distributor forms a full inverse subsemigroup.

Conditions are identified when solutions form strong semilattices.

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.