On the passage from finite braces to pre-Lie rings

TL;DR

This paper establishes a correspondence between strongly nilpotent braces and nilpotent pre-Lie rings of certain sizes, providing a new link between these algebraic structures for large primes.

Contribution

It introduces a bijective correspondence between strongly nilpotent braces and nilpotent pre-Lie rings, and an injective map from pre-Lie rings to braces under specific conditions.

Findings

One-to-one correspondence for large primes

Injective mapping for certain nilpotent structures

Extension of methods from previous work [41]

Abstract

Let p be a prime number. We show that there is a one-to-one correspondence between the set of strongly nilpotent braces and the set of nilpotent pre-Lie rings of cardinality $p^{n}$, for sufficiently large p. Moreover, there is an injective mapping from the set of left nilpotent pre-Lie rings into the set of left nilpotent braces of cardinality $p^{n}$ for n+1<p. For the passage from pre-Lie rings to braces we use exactly the same method as suggested in [41].