Affine structures on groups and semi-braces

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.

arXiv comment23 pages

References in corpus (15)

• Set-theoretical solutions to the quantum Yang-Baxter equation
• On set-theoretical solutions of the quantum Yang-Baxter equation
• Skew braces and the Yang-Baxter equation
• On skew braces (with an appendix by N. Byott and L. Vendramin)
• Skew left braces of nilpotent type
• Factorizations of skew braces
• Bi-Skew Braces and Regular Subgroups of the Holomorph
• Bi-skew braces and Hopf Galois structures
• Skew braces of size $pq$
• From endomorphisms to bi-skew braces, regular subgroups, the Yang--Baxter equation, and Hopf--Galois structures
• On bi-skew braces and brace blocks
• Set-theoretic solutions of the Yang-Baxter equation associated to weak braces
• Brace blocks from bilinear maps and liftings of endomorphisms
• Nilpotency in left semi-braces
• Inverse semi-braces and the Yang-Baxter equation