Markov and Artin Normal Form Theorem for Braid Groups

Bokut, L. A.; Chaynikov, V. V.; Shum, K. P.

doi:10.1142/s0218196707003950

Cited by 12 publications

(19 citation statements)

References 7 publications

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“…5 Normal forms for groups and semigroups -Braid groups in Artin-Burau generators (Bokut-Chanikov-Shum [9]); in Artin-Garside generators (Bokut [7]); in Birman-Ko-Lee generators (Bokut [8]); in Adyan-Thurston generators (Chen-Zhong [56]).…”

Section: 8mentioning

confidence: 99%

“…Recently some new Composition-Diamond lemmas are given: for free algebra k Y ⊗ k X [13], for Lie algebras over commutative algebras [14], for metabelian Lie algebras [42], for semirings [24], for Rota-Baxter algebras [16], for L-algebras [17], for Vinberg-Koszul-Gerstenhaber right-symmetric algebras [18], for categories [20], for dialgebras [22], for associative algebras with multiple operations [25], for associative n-conformal algebras [27], for associative conformal algebras [29], for Lie superalgebras [31], for differential algebras [43], for λ-differential associative algebras with multiple operators [60], for commutative algebras with multiple operators and free commutative Rota-Baxter algebras [61], etc. By using the above and the known Composition-Diamond lemmas, some applications are obtained: for embeddings of algebras [23,52], for free inverse semigroups [28], for conformal algebras [30], for relative Gröbner-Shirshov bases of algebras and groups [36], for extensions of groups and algebras [40,41], for some word problems [10,44], for some Lie algebras [47], for partially commutative Lie algebras [50,59], for braid groups [7,8,9,48,56], for PBW theorems [14,18,22,35,45,46,…”

Section: Introductionmentioning

confidence: 99%

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Gröbner–Shirshov bases and their calculation

2014

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In this survey we give an exposition of the theory of Gröbner-Shirshov bases for associative algebras, Lie algebras, groups, semigroups, -algebras, operads, etc. We mention some new Composition-Diamond lemmas and applications.Keywords Gröbner basis · Gröbner-Shirshov basis · Composition-Diamond lemma · Congruence · Normal form · Braid group · Free semigroup · Chinese monoid · Plactic monoid · Associative algebra · Lie algebra · Lyndon-Shirshov basis · Lyndon-Shirshov word · PBW theorem · -algebra · Dialgebra · Semiring · Pre-Lie algebra · Rota-Baxter algebra · Category · ModuleSupported by the NNSF of China (11171118) The set of all associative Lyndon-Shirshov words in X NLSW(X)The set of all non-associative Lyndon-Shirshov words in X PBW theoremThe Poincare-Birkhoff-Witt theorem X * The free monoid generated byThe free commutative monoid generated by X X * *The set of all non-associative words (u) in X gp X |S The group generated by X with defining relations S sgp X |S The semigroup generated by X with defining relations S k A field K A commutative algebra over k with unity k X The free associative algebra over k generated by X k X |S The associative algebra over k with generators X and defining relations S S c A Gröbner-Shirshov completion of S I d(S)The ideal generated by a set S sThe maximal word of a polynomial s with respect to some ordering < Irr(S)The set of all monomials avoiding the subwords for all s ∈ S k [X ] The polynomial algebra over k generated by X Lie(X )The free Lie algebra over k generated by X Lie K (X )The free Lie algebra generated by X over a commutative algebra K

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Section: 8mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gröbner–Shirshov bases and their calculation

2014

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“…Bokut-Chainikov-Shum [10] found a Gröbner-Shirshov basis for B n+1 in Artin-Burau generators and as a corollary the Markov-Artin normal form is followed. Bokut-Fong-Ke-Shiao [11] found a Gröbner-Shirshov basis for the braid semigroup…”

Section: Introductionmentioning

confidence: 94%

Gröbner-Shirshov Bases for Braid Groups in Adyan-Thurston Generators

Chen

Zhong

2013

Algebra Colloq.

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“…Let us refer, for example, to [3][4][5][6][7][8][9] for preliminaries on Gröbner-Shirshov bases and examples of recent related results.…”

Section: Preliminariesmentioning

confidence: 99%

Internet Security Applications of Gröbner-Shirshov Bases

Kelarev

Yearwood

Watters

2010

Asian-European J. Math.

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This article is motivated by internet security applications of multiple classifiers designed for the detection of malware. Following a standard approach in data mining, Dazeley et al. (Asian-European J. Math. 2 (2009)(1) 41-56) used Gröbner-Shirshov bases to define a family of multiple classifiers and develop an algorithm optimizing their properties. The present article complements and strengthens these results. We consider a broader construction of classifiers and develop a new and more general algorithm for the optimization *

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Markov and Artin Normal Form Theorem for Braid Groups

Abstract: In this paper we will present the results of Artin–Markov on braid groups by using the Gröbner–Shirshov basis. As a consequence we can reobtain the normal form of Artin–Markov–Ivanovsky as an easy corollary.

Cited by 12 publications

References 7 publications

Gröbner–Shirshov bases and their calculation

Gröbner–Shirshov bases and their calculation

Gröbner-Shirshov Bases for Braid Groups in Adyan-Thurston Generators

Internet Security Applications of Gröbner-Shirshov Bases

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