Multifraction reduction I: The 3-Ore case and Artin–Tits groups of type FC

Dehornoy, Patrick

doi:10.4171/jca/1-2-3

Cited by 11 publications

(42 citation statements)

References 33 publications

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“…The results of the current paper leaves the question of whether Artin-Tits monoids of sufficiently large type satisfy Conjecture A open, but, at least, we know now that R-reduction is relevant for them. In view of this result, and those of [2], which settle the case of type FC, the "first" case for which nothing is known in terms of reduction (nor of the word problem) is the Artin-Tits monoid with exponents 3, 3, 3, 3, 3, 2, that is, the monoid a, b, c, d | aba = bab, aca = cac, bcb = cbc, ada = dad, bdb = dbd, cd = dc + . We hope that further progress will arise soon…”

Section: 3supporting

confidence: 60%

“…If M is a gcd-monoid, a family R M of rewrite rules on F M is defined in [2]: for a, b in F M , and for i 1 and x ∈ M , we write a • R i,x = b if we have b = a , b k = a k for k = i − 1, i, i + 1, and there exists x ′ satisfying for i even:…”

Section: Padded Multifraction Reductionmentioning

confidence: 99%

“…In the definition of multifraction reduction used in [2,3,6] and repeated in Section 1.2, in order for a • R i,x to be defined, x and a i have to admit a common multiple, a left-or right-multiple depending on the parity of i. In order to define a variant of this reduction, we now relax the assumption and only require that x and some divisor of a i admit a common multiple; then, we again remove x from a i+1 and push it to the left or right side using the lcm operation, this time through the divisor of a i that is involved and not necessarily through the whole of a i .…”

Section: Padded Multifraction Reductionmentioning

confidence: 99%

“…By the results of [2], Conjecture A padded is true for every Artin-Tits monoid of type FC. Here we prove: Theorem 1.…”

mentioning

confidence: 95%

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Multifraction reduction IV: Padding and Artin–Tits monoids of sufficiently large type

Dehornoy

Holt

Rees

2018

Journal of Pure and Applied Algebra

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We investigate the padded version of reduction, an extension of multifraction reduction as defined in arXiv:1606.08991, and connect it both with ordinary reduction and with the so-called Property H. As an application, we show that all Artin-Tits groups of sufficiently large type satisfy some weakening Conjecture A padded of Conjecture A, thus showing that the reduction approach is relevant for these groups.Reduction of multifractions, which was introduced in [2] and [3], is a new approach to the word problem for Artin-Tits groups and, more generally, for groups that are enveloping groups of monoids in which the divisibility relations have weak lattice properties ("gcd-monoids"). It is based on a rewrite system ("R-reduction") that extends the usual free reduction for free groups, as well as the rewrite systems known for Artin-Tits groups of spherical type, and more generally Garside groups. It was proved in [2] that R-reduction is convergent for all Artin-Tits groups of type FC, and in [3] that a certain condition called semi-convergence, weaker than convergence, is sufficient to obtain the decidability of the word problem, leading to the main conjecture ("Conjecture A") that R-reduction is semi-convergent for every Artin-Tits monoid.The aim of the current paper is to exploit the observation that semi-convergence up to Turing-computable padding, a weakening of semi-convergence, is again sufficient to solve the word problem. By padding, we mean the insertion of an even number of trivial components at the beginning of a multifraction.The main results we prove are as follows. First, we have a simple criterion for the word problem: Proposition 1.6. If M is a strongly noetherian gcd-monoid with finitely many basic elements, for which R-reduction is semi-convergent up to f -padding for some Turing-computable map f , then the word problem of U(M ) is decidable.Next, we establish a simple connection between the padded version of semiconvergence, the semi-convergence of a variant of R-reduction ("split reduction" or "S-reduction") and Property H of [1,4,8]:Proposition 1.14. If M is a gcd-monoid and (S, R) is an lcm-presentation for M , then the following are equivalent:(i) R-reduction is semi-convergent for M up to padding;

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Section: 3supporting

confidence: 60%

Section: Padded Multifraction Reductionmentioning

confidence: 99%

Section: Padded Multifraction Reductionmentioning

confidence: 99%

“…By the results of [2], Conjecture A padded is true for every Artin-Tits monoid of type FC. Here we prove: Theorem 1.…”

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confidence: 95%

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Multifraction reduction IV: Padding and Artin–Tits monoids of sufficiently large type

Dehornoy

Holt

Rees

2018

Journal of Pure and Applied Algebra

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“…However, for #C 2, the monoid B + + + n,C admits no common multiple: for a = b, the elements σ 1 ). In [9], the embeddability criterion of Ore's theorem is extended to cancellative monoids with no nontrivial invertible elements that satisfy the following "3-Ore condition":…”

Section: Applications To Variants Of Braid Monoidsmentioning

confidence: 99%

A cancellativity criterion for presented monoids

Dehornoy

2018

Semigroup Forum

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We establish a new, fairly general cancellativity criterion for a presented monoid that properly extends the previously known related criteria. It is based on a new version of the word transformation called factor reversing, and its specificity is to avoid any restriction on the number of relations in the presentation. As an application, we deduce the cancellativity of some natural extension of Artin's braid monoid in which crossings are colored.Establishing that a presented monoid (or semigroup) is cancellative is in general a nontrivial task, for which not so many methods are known [12, sec. 5.3]. If a distinguished expression ("normal form") has been identified for each element of the monoid, and if, for each element a of the considered monoid M and every generator s of the considered presentation, the normal form of a can be retrieved from that of sa and s, then one can indeed conclude that sa = sb implies a = b. But, when no normal form is known, no generic method is available. Adjan's criterion based on the left graph [1,14] is useful, but, by definition, it applies only to presentations with (very) few defining relations. Ultimately relying on Garside's analysis of the braid monoids B + + + n [11], the so-called reversing method [5,8] provides a simple criterion, which proved to be useful for many concrete presentations, typically those of all Artin-Tits monoids. However, an intrinsic limitation of the method is that it only applies to monoid presentations (S, R) that contain a limited number of relations, namely those such that, for all s, t in S, there exists at most one relation of the form s... = t... in R ("right-complemented" presentations). The aim of this paper is to extend the previous criterion by developing a new approach that requires no limitation on the number of defining relations. The result we prove takes the following form:Proposition. Assume that (S, R) is a monoid presentation such that (i) there exists an ≡ R -invariant map λ from S * to ordinals satisfying λ(sw) > λ(w) for all s in S and w in S * , and (ii) for every s in S, for every relation w = w ′ in R, and for every (S, R)-grid from (s, w), there exists an equivalent grid from (s, w ′ ), and vice versa, and (iii) there is no relation sw = sw ′ in R with w, w ′ distinct. Then the monoid associated with (S, R) admits left cancellation.In the above statement, ≡ R refers to the congruence on the free monoid S * generated by the relations of R, and an (S, R)-grid is a certain type of rectangular van Kampen diagram specified in Definition 1.1 below. Note that Condition (i) in the above statement is trivial when each relation in R consists of two words with 1991 Mathematics Subject Classification. 20M05, 20M12, 20F36.

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Gcd-monoids arising from homotopy groupoids

Wehrung

2018

Semigroup Forum

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The interval monoid Υ(P ) of a poset P is defined by generators [x, y], where x ≤ y in P , and relations [x, x] It embeds into its universal group Υ ± (P ), the interval group of P , which is also the universal group of the homotopy groupoid of the chain complex of P . We prove the following results:• The monoid Υ(P ) has finite left and right greatest common divisors of pairs (we say that it is a gcd-monoid) iff every principal ideal (resp., filter) of P is a join-semilattice (resp., a meet-semilattice). • For every group G, there is a connected poset P of height 2 such that Υ(P )is a gcd-monoid and G is a free factor of Υ ± (P ) by a free group. Moreover, P can be taken to be finite iff G is finitely presented. • For every finite poset P , the monoid Υ(P ) can be embedded into a free monoid. • Some of the results above, and many related ones, can be extended from interval monoids to the universal monoid Umon(S) of any category S. This enables us, in particular, to characterize the embeddability of Umon(S) into a group, by stating that it holds at the hom-set level.We thus obtain new easily verified sufficient conditions for embeddability of a monoid into a group. We illustrate our results by various examples and counterexamples.

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Multifraction reduction I: The 3-Ore case and Artin–Tits groups of type FC

Cited by 11 publications

References 33 publications

Multifraction reduction IV: Padding and Artin–Tits monoids of sufficiently large type

Multifraction reduction IV: Padding and Artin–Tits monoids of sufficiently large type

A cancellativity criterion for presented monoids

Gcd-monoids arising from homotopy groupoids

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