Post Clifford semigroups, the Yang-Baxter equation, relative Rota--Baxter Clifford semigroups and dual weak left braces

Abstract

As generalizations of Rota--Baxter groups, Rota--Baxter Clifford semigroups have been introduced by Catino, Mazzotta and Stefanelli in 2023. Based on their pioneering results, in this paper we first continue to study Rota--Baxter Clifford semigroups. Inspired by the corresponding results in Rota--Baxter groups, we firstly obtain some properties and construction methods for Rota--Baxter Clifford semigroups, and then study the substructures and quotient structures of these semigroups. On the other hand, as generalizations of post-groups, Rota--Baxter Clifford semigroups and braided groups, in this paper we introduce and investigate post Clifford semigroups, relative Rota--Baxter Clifford semigroups and braided Clifford semigroups, respectively. We prove that the categories of strong post Clifford semigroups, dual weak left braces, bijective strong relative Rota-Baxter Clifford semigroups and braided Clifford semigroups are mutually pairwise equivalent, and the category of post Clifford semigroups is equivalent to the category of bijective relative Rota--Baxter Clifford semigroups, respectively. As a consequence, we prove that both post Clifford semigroups, relative Rota--Baxter Clifford semigroups and braided Clifford semigroups can provide set-theoretical solutions for the Yang--Baxter equation. The substructures and quotient structures of relative Rota--Baxter Clifford semigroups are also considered.