Some characterizations of weak left braces

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, $\lambda$-homomorphic and $\lambda$-anti-homomorphic weak left braces are introduced and the algebraic structures of these weak left braces are given.