Dual weak braces and Płonka sums of solutions of the Yang–Baxter equation
Tom 129 / 2025
Streszczenie
We examine the algebraic structure of dual weak braces and illustrate how they produce set-theoretical solutions of the Yang–Baxter equation (Catino et al. 2022, 2023). In particular, we focus on two results of Catino et al. (2023). The first one is a complete description of dual weak braces that is strictly connected to the construction of strong semilattice of groups which characterizes Clifford semigroups, also known as Płonka sums of groups. As a consequence, we show that dual weak braces are Płonka sums of skew braces. The second one is that solutions connected with dual weak braces have a behaviour close to being bijective and non-degenerate and, more specifically, that they are strong semilattices of bijective solutions. In this regard, we deal with the notion of Płonka sums of solutions, prove it is equivalent to that of strong semilattice of solutions, and show that solutions associated to dual weak braces are Płonka sums of bijective and non-degenerate solutions.
