Rewriting Systems and Embedding of Monoids in Groups

Chouraqui, Fabienne

doi:10.1515/gcc.2009.131

Cited by 3 publications

(4 citation statements)

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“…Furthermore, it is a consequence of a more general due to Paris [145] that any trace monoid embeds in its corresponding right-angled Artin group; this is also easy to prove using the theory of rewriting systems, cf. Chouraqui [41]. Right-angled Artin groups play a key rôle in modern geometric group theory due to their rich subgroup structure, as demonstrated in the work of Wise and Haglund on special cube complexes (an area far beyond the scope of this survey; the reader is directed to Wise's monograph [178]) and the recent interest in solving equations over right-angled Artin groups, cf.…”

Section: Example 24mentioning

confidence: 99%

The word problem for one-relation monoids: a survey

Brodda

2021

Semigroup Forum

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This survey is intended to provide an overview of one of the oldest and most celebrated open problems in combinatorial algebra: the word problem for one-relation monoids. We provide a history of the problem starting in 1914, and give a detailed overview of the proofs of central results, especially those due to Adian and his student Oganesian. After showing how to reduce the problem to the left cancellative case, the second half of the survey focuses on various methods for solving partial cases in this family. We finish with some modern and very recent results pertaining to this problem, including a link to the Collatz conjecture. Along the way, we emphasise and address a number of incorrect and inaccurate statements that have appeared in the literature over the years. We also fill a gap in the proof of a theorem linking special inverse monoids to one-relation monoids, and slightly strengthen the statement of this theorem.

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Section: Example 24mentioning

confidence: 99%

The word problem for one-relation monoids: a survey

Brodda

2021

Semigroup Forum

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“…All three questions have affirmative answers in the case of right-angled Artin groups, i.e. when M (Γ) is a binary matrix: Paris [70] proved that every right-angled Artin monoid (defined analogously) embeds in its corresponding right-angled Artin group (this can also be deduced in a simpler manner via rewriting systems [15]); Green [33] solved the word problem via normal forms; and Hermiller & Meier [36] constructed finite complete rewriting systems for right-angled Artin groups which additionally lead to an automatic structure.…”

Section: Warmup -Torus Knot Groupsmentioning

confidence: 99%

“…Proof. We will let G = BS (p) (1, pq), with presentation as in (15), and let A = {a, b}. If p = 1, BS (p) (1, pq) = BS(1, q) ∼ = Gp x 1 , x 2 | x 1 = (x q 1 ) x2 = NG(1, q) = NG(1, pq), as required.…”

Section: Torsion Subgroupsmentioning

confidence: 99%

On the Diophantine problem in some one-relator groups

Nyberg-Brodda¹

2022

Preprint

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We study the Diophantine problem, i.e. the decision problem of solving systems of equations, for some families of one-relator groups, and provide some background for why this problem is of interest. The method used is primarily the Reidemeister-Schreier method, together with general recent results by Dahmani & Guirardel and Ciobanu, Holt & Rees on the decidability of the Diophantine problem in general classes of groups. First, we give a sample of the methods of the article by proving that the one-relator group with defining relation a m b n = 1 is virtually a direct product of hyperbolic groups for all m, n ≥ 0, and thus conclude decidability of the Diophantine problem in such groups. As a corollary, we obtain that the Diophantine problem is decidable in any torus knot group. Second, we study the twogenerator, one-relator groups Gm,n with defining relation a commutator [a m , b n ] = 1, where m, n ≥ 1. In doing so, we define and study a natural class of groups (rabsags), related to right-angled Artin groups (raags). We reduce the Diophantine problem in the groups Gm,n to the Diophantine problem in groups which are virtually certain rabsags. As a corollary of our methods, we show that the submonoid membership problem is undecidable in the group G 2,2 with the single defining relation [a 2 , b 2 ] = 1. We use the recent classification by Gray & Howie of raag subgroups of one-relator groups to classify the raag subgroups of some rabsags, showing the potential usefulness of one-relator theory to this area. Finally, we define and study Newman groups NG(p, q), which are (p + 1)-generated one-relator groups generalising the solvable Baumslag-Solitar groups. We show that all such groups are hyperbolic, and thereby also conclude decidability of their Diophantine problem.

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“…In group, theory rewriting gives algorithmic methods for decision problems, such as the word/conjugacy/geodesic problems (Chouraqui, 2009(Chouraqui, , 2011Diekert et al, 2012Diekert et al, , 2010Le Chenadec, 1984, 1986. In most cases, the method consists in constructing a convergent presentation of the considered group.…”

Section: Rewriting In Groupsmentioning

confidence: 99%

Confluence of algebraic rewriting systems

Chenavier

Dupont

Malbos

2021

Math. Struct. Comp. Sci.

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Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical branching lemma that proves local confluence from confluence of the critical branchings. The analysis of local confluence of rewriting systems on algebraic structures, such as groups or linear algebras, is complicated because of the underlying algebraic axioms. This article introduces the structure of algebraic polygraph modulo that formalizes the interaction between the rules of an algebraic rewriting system and the inherent algebraic axioms, and we show a critical branching lemma for algebraic polygraphs. We deduce a critical branching lemma for rewriting systems on algebraic models whose axioms are specified by convergent modulo rewriting systems. We illustrate our constructions for string, linear, and group rewriting systems.

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Rewriting Systems and Embedding of Monoids in Groups

Cited by 3 publications

References 11 publications

The word problem for one-relation monoids: a survey

The word problem for one-relation monoids: a survey

On the Diophantine problem in some one-relator groups

Confluence of algebraic rewriting systems

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