Associativity of the ⋆-product of skew braces
Abstract
For braces, associativity of the ⋆-product is equivalent to two-sidedness. We extend this result to skew braces by proving that any two of the following conditions imply the third: two-sidedness, associativity of the ⋆-product, and centrality of A⋆ A in the additive group. We then study two-sided skew braces with associative ⋆-product through the radical ring associated with their additive abelianization. This yields a canonical fibre-product description modulo an annihilator ideal, exact sequences relating higher ⋆-products to the powers of the radical ring, coincidence of the left, right and strong ⋆-series, and sharp bounds on their nilpotency indices. We also obtain an explicit description of the lower central series, classify the structures whose additive abelianization is infinite cyclic, and characterize when they are bi-skew.
Related
- Classification of skew left braces with additive group isomorphic to the infinite dihedral group
- Determinant characters and multiplicative solubility of finite skew braces
- Bisoluble and Binilpotent Skew Braces of Finite Morley Rank
- Skew braces whose left series is not generated by right-normed star products
- Classification of finite left-simple skew braces
BibTeX
@misc{sciandra2026associativity,
title = {{Associativity of the $\star$-product of skew braces}},
author = {Andrea Sciandra},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22995},
eprint = {2609.22995},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22995},
}
RIS
TY - UNPB
TI - Associativity of the ⋆-product of skew braces
AU - Andrea Sciandra
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22995
UR - https://arxiv.org/abs/2609.22995
AB - For braces, associativity of the ⋆-product is equivalent to two-sidedness. We extend this result to skew braces by proving that any two of the following conditions imply the third: two-sidedness, associativity of the ⋆-product, and centrality of A⋆ A in the additive group. We then study two-sided skew braces with associative ⋆-product through the radical ring associated with their additive abelianization. This yields a canonical fibre-product description modulo an annihilator ideal, exact sequences relating higher ⋆-products to the powers of the radical ring, coincidence of the left, right and strong ⋆-series, and sharp bounds on their nilpotency indices. We also obtain an explicit description of the lower central series, classify the structures whose additive abelianization is infinite cyclic, and characterize when they are bi-skew.
ER -