相关论文: On the passage from finite braces to pre-Lie rings

We classify nilpotent pre-Lie rings of cardinality $p^4$ and thereby braces of the same cardinality, for a sufficiently large prime $p$. It has been shown that nilpotent pre-Lie rings of cardinality $p^n$ correspond to strongly nilpotent…

Let $A$ be a brace of cardinality $p^{n}$ where $p>n+1$ is prime, and let $ann (p^{2})$ be the set of elements of additive order at most $p^{2}$ in this brace. We construct a pre-Lie ring related to the brace $A/ann(p^{2})$. In the case of…

Let A be a brace of cardinality p^{n} where p>n+1 is prime, and ann(p^{i}) be the set of elements of additive order at most p^{i} in this brace. A pre-Lie ring related to the brace A/ann(p^{2}) was constructed in [8]. We show that there is…

Let A be a brace of cardinality $p^{n}$ for some prime number $p$. Suppose that either (i) the additive group of brace $A$ has rank smaller than $p-3$, or (ii) $A^{\frac {p-1}2}\subseteq pA$ or (iii) $p^{i}A$ is an ideal in in $A$ for each…

In this article the right nilpotent $\mathbb{F}_p$-braces of cardinality $p^5$ has been classified. We use the connection between nilpotent $\mathbb{F}_p$-braces of cardinality $p^5$ and nilpotent pre-Lie algebras of the same order,…

Let $p>3$ be a prime number and let $A$ be a brace whose additive group is a direct sum of cyclic groups of cardinalities larger than $p^{\alpha }$ for some $\alpha $. Suppose that either (i) $A^{\lfloor{\frac {p-1}4}\rfloor}\subseteq pA$…

In this paper a simple algebraic formula is obtained for the correspondence between finite right nilpotent Fp-braces and finite nilpotent pre-Lie algebras. This correspondence agrees with the correspondence using Lazard's correspondence…

We develop a Lazard correspondence between post-Lie rings and skew braces that satisfy a natural completeness condition. This is done through a thorough study of how the Lazard correspondence behaves on semi-direct sums of Lie rings. In…

In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right R-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors,…

We describe all $\mathbb F_{p}$-braces of cardinality $p^{4}$ which are not right nilpotent. Our solution illustrates a general way of investigating $\mathbb F_{p}$-braces of cardinality $p^{n}$ with a given multiplicative group. The…

For a prime p and natural number n with p greater than or equal to n, we establish the existence of a non-functorial one-to-one correspondence between isomorphism classes of groups of order p^n whose derived subgroup has exponent dividing…

We classify the nilpotent Lie rings of order $p^8$ with maximal class for $p \ge 5$. This also provides a classification of the groups of order $p^8$ with maximal class for $p \ge 11$ via the Lazard correspondence.

The Lazard correspondence induces a close relation between the $p$-groups of maximal class and a certain type of Lie ring constructed from $p$-adic number fields. Our aim here is to investigate such Lie rings. In particular, we show that…

We determine right nilpotency of braces of cardinality $p^4$. If a brace of cardinality $p^4$ has an abelian multiplicative group, then it is left and right nilpotent, so we only consider braces with non-abelian multiplicative groups. We…

We introduce left and right series of left semi-braces. This allows to define left and right nilpotent left semi-braces. We study the structure of such semi-braces and generalize some results, known for skew left braces, to left…

We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies a given polynomial identity $r(t)=0$. For the polynomial $r=t^n-1$ we obtain results on the nilpotency of Lie algebras admitting a…

This article provides a detailed description of some nilpotent left braces generated by one element.

We construct uncountably categorical 3-nilpotent groups of exponent p > 3. They are not one-based and do not allow the interpretation of an infinite field. Therefore they are counterexamples to Zilbers Conjecture. First 2-nilpotent new…

This paper aims to introduce the concept of nilpotency and capability in multiplicative Lie algebras. Also, we see the existence of covers of a multiplicative Lie algebra and thoroughly examine their relationships with capable and perfect…

We study some relations between left cancellative left semi-braces and other existing algebraic structures. In particular, we show that every left semi-brace arises from a left seminear-ring, extending the correspondence given by Rump…