Some characterizations of weak left braces

TL;DR

This paper characterizes weak left braces, a generalization of skew left braces, using algebraic structures like Gamma functions and inverse subsemigroups, and introduces various subclasses with specific properties.

Contribution

It introduces new notions such as good inverse subsemigroups and Gamma functions for Clifford semigroups, providing a framework to characterize weak left braces.

Findings

Weak left braces are characterized via Gamma functions and inverse subsemigroups.

Introduces symmetric, λ-homomorphic, and λ-anti-homomorphic weak left braces.

Provides algebraic structures and classifications of these 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.