Deformed solutions of the Yang-Baxter equation associated to dual weak left $\star$-braces

Abstract

As generalizations of dual weak left braces and skew left braces, in this paper, dual weak left $\star$-braces and square skew left braces are introduced, respectively. We firstly show that a dual weak left $\star$-brace is exactly a strong semilattice of a family of square skew left braces. Then we introduce distributors for dual weak left $\star$-braces and prove that the map deformed by each distributor is always a solution of the Yang-Baxter equation. Our work may be regarded as extending and enriching some related results on skew left braces and weak left braces in literature.