Affine structures on groups and semi-braces

TL;DR

This paper introduces affine structures on groups, establishing an equivalence with semi-braces, and provides methods to construct and analyze complex semi-brace examples, expanding the understanding of algebraic structures related to solutions of the Yang-Baxter equation.

Contribution

It presents a new categorical framework for semi-braces via affine structures, including methods for constructing bi-skew braces and semi-braces not arising from matched products.

Findings

Affine structures form a category equivalent to semi-braces.

Constructed examples of bi-skew braces, including non-λ-homomorphic ones.

Provided a method to build semi-braces from Zappa products of groups.

Abstract

We introduce affine structures on groups and show they form a category equivalent to that of semi-braces. In particular, such a new description of semi-braces includes that presented by Rump for braces. By specific affine structures, we provide several instances of bi-skew braces, including some that are not $\lambda$-homomorphic. Finally, we give a method for determining affine structures on the Zappa product of two groups both endowed with affine structures and prove that such a construction allows for obtaining semi-braces that are not matched product of semi-braces.