Some characterizations of weak left braces
Shoufeng Wang
TL;DR
The paper develops a comprehensive framework linking weak left braces to inverse semigroups, Clifford semigroups, Gamma functions, and affine structures to study (degenerate) set-theoretical solutions of the Yang–Baxter equation. It proves that weak left braces correspond to good inverse subsemigroups of ${\rm End\,(S,+)}\rtimes (S,+)$ and that Gamma/dual-Gamma functions classify them, while affine structures offer constructive methods. It further extends the theory to symmetric, $\lambda$-homomorphic, and $\lambda$-anti-homomorphic weak left braces, showing these variants are dual weak left braces and can be built as strong semilattices of skew left braces. Together, these results provide systematic tools for generating and analyzing brace-structured solutions and broaden the connections between brace theory and semigroup theory.
Abstract
As generalizations of skew left braces, weak left braces were introduced recently by Catino, Mazzotta, Miccoli and Stefanelli to study ceratin special degenerate set-theoretical solutions of the Yang-Baxter equation. In this note, as analogues of the notions of regular subgroups of holomorph of groups, Gamma functions on groups and affine and semi-affine structures on groups, we propose the notions of good inverse subsemigroups and Gamma functions associated to Clifford semigroups and affine structures on inverse semigroups, respectively, by which weak left braces are characterized. Moreover, symmetric, $λ$-homomorphic and $λ$-anti-homomorphic weak left braces are introduced and the algebraic structures of these weak left braces are given.