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From pre-Lie rings back to braces
Rings and Algebras · Submitted 4 Aug 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that there is a formula dependent only on the additive group of the brace A which reverses the construction from [8].
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The abstract, sentence by sentence
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 a formula dependent only on the additive group of the brace A which reverses the construction from [8]. As an application example it is shown that the brace A/ann(p^{4}) is the group of flows of a left nilpotent pre-Lie algebra.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.99 | 0.16 | 0.03 | asserted 0.65 |
| s1 | A pre-Lie ring related to the brace A/ann(p^{2}) was constructed in [8]. | background 1.00 | 0.06 | 0.04 | asserted 1.00 |
| s2 | We show that there is a formula dependent only on the additive group of the brace A which reverses the construction from [8]. | contribution 0.60 | 0.95 | 0.81 | asserted 1.00 |
| s3 | As an application example it is shown that the brace A/ann(p^{4}) is the group of flows of a left nilpotent pre-Lie algebra. | result 0.95 | 0.90 | 0.14 | asserted 1.00 |
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The paper
Original abstract
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 a formula dependent only on the additive group of the brace A which reverses the construction from [8]. As an application example it is shown that the brace A/ann(p^{4}) is the group of flows of a left nilpotent pre-Lie algebra.
Human review
How the literature uses it
3 arXiv papers cite this in 6 sentences
Papers citing it · 3 sentences without a confident label
Uses its method or tool 1 paper
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By putting $a^∘p^k$ and $b^∘k$ at $j$th place instead of the first place (similarly as it was done in unpublished paper [ref] , on page $14$), and by applying the same reasoning we obtain [equation] for any $x_i, x_i'∈{a,b}$ for this given $t$.
On pre-Lie rings related to some non-Lazard braces · Function $f$ · 0.94
Cites it as background 1 paper
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Arrangement: The paper is structured as follows: Section $2$ provides preliminaries discussing the established connection between braces and pre-Lie algebras by Smoktunowicz [ref] .
Classification of right nilpotent $\mathbb{F}_p$-braces of cardinality $p^5$ · Introduction · 0.96
What it cites · 18 sentences without a confident label
Uses the method or tool of 2 papers
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Let $(A/ann (p^2), +, ∙)$ be the corresponding pre-Lie ring, constructed as in [ref] from the brace $(A, +, ∘)$ (notice that this construction is quoted in Theorem [ref]).
From braces to pre-Lie rings · Braces $A/ann(p^4)$ and groups of flows · 0.90
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We will use the same notations for the construction of the group of flows as in [ref] .
On the passage from finite braces to pre-Lie rings · Braces $A/ann(p^4)$ and groups of flows · 0.81
Compares with 1 paper
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We will use the same notation as in [ref] .
From braces to pre-Lie rings · Introduction · 0.83
Cites as support 1 paper
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Note that, by Lemma $17$, [ref] , $ann(p^i)$ is an ideal in the above brace $A$.
On the passage from finite braces to pre-Lie rings · Introduction · 0.97
Cites as background 2 papers
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For the background section we refer the reader to the background section of [ref] .
From braces to pre-Lie rings · Introduction · 1.00
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For more details, see [ref] .
On the passage from finite braces to pre-Lie rings · Introduction · 0.97
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