Primitive and Almost Primitive Elements of Schreier Varieties

Артамонов, В. А.; Klimakov, A. V.; Михалев, А. В.; Mikhalëv, A. V.

doi:10.1007/s10958-019-4148-2

Cited by 4 publications

(2 citation statements)

References 130 publications

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“…We wish to dedicate this paper to the memory of Vyacheslav Alexandrovich Artamonov who passed away in June 2021. Not only had he been interested in Nielsen-Schreier varieties of algebras throughout his mathematical life [7,8,9,10], but also his mathematical heritage is strongly connected with the operad theory: while operads were first defined by J. P. May in 1971 in his work on iterated loop spaces [37,49], the same notion seems to have been first introduced under a much more technical name of a "clone of multilinear operations" in Artamonov's 1969 paper [6].…”

Section: Acknowledgementsmentioning

confidence: 99%

An effective criterion for Nielsen-Schreier varieties

Dotsenko¹,

Umirbaev²

2022

Preprint

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All algebras of a certain type are said to form a Nielsen-Schreier variety if every subalgebra of every free algebra is free. Using methods of the operad theory, we propose an effective combinatorial criterion for that property in the case of algebras over a field of zero characteristic. Using this criterion, we show that the variety of all pre-Lie algebras is Nielsen-Schreier, and that, quite surprisingly, there are already infinitely many Nielsen-Schreier varieties of algebras with one binary operation and identities of degree three.

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Section: Acknowledgementsmentioning

confidence: 99%

An effective criterion for Nielsen-Schreier varieties

Dotsenko¹,

Umirbaev²

2022

Preprint

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“…This concept generalizes the relationship between the variety of associative algebras and the variety of Lie algebras described by the Poincare-Birkhoff-Witt theorem [4,5]. The study of primitive elements of a free algebra, which are elements contained in some free generating set for the free algebra, is also a subject of current research [6,7].…”

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confidence: 93%

Embeddings of Free Magmas with Applications to the Study of Free Non-Associative Algebras

Ismail¹

2018

Preprint

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We introduce an embedding of the free magma on a set A into the direct product of the free magma on a singleton set and the free semigroup on A. This embedding is then used to prove several theorems related to algebraic independence of subsets of the free non-associative algebra on A. Among these theorems is a generalization of a result due to Kurosh, which states that every subalgebra of a free non-associative algebra is also free.

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Algebras with Single Defining Relation

Mikhalev

2024

J Math Sci

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Primitive and Almost Primitive Elements of Schreier Varieties

Cited by 4 publications

References 130 publications

An effective criterion for Nielsen-Schreier varieties

An effective criterion for Nielsen-Schreier varieties

Embeddings of Free Magmas with Applications to the Study of Free Non-Associative Algebras

Algebras with Single Defining Relation

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