Relative Rota--Baxter Clifford semigroups and dual weak left braces

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, we continue to study Rota--Baxter Clifford semigroups and their generalizations in the present paper. 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, we introduce the notion of relative Rota--Baxter Clifford semigroups and explore the relationship between these semigroups and dual weak left braces, which are known to give set-theoretical solutions of the Yang--Baxter equation. More specifically, we show that every relative Rota--Baxter Clifford semigroup gives rise to a dual weak left brace, and conversely, every dual weak left brace can be obtained from a bijective strong relative Rota--Baxter Clifford semigroup. Furthermore, it turns out that there is an equivalence between the two categories of bijective strong Rota--Baxter Clifford semigroups and dual weak left braces. The substructures and quotient structures of Rota--Baxter Clifford semigroups are also considered.