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BRAID GROUPS, CLUSTERS, AND FREE PROBABILITY: AN OUTLINE FROM THE AIM WORKSHOP, JANUARY 2005

Number theory · math.NT · math.PR · posed by Drew Armstrong · open

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Statement

Given a cancellative, finitely-generated monoid M in which lcm's exist, is M necessarily a Garside monoid? That is, does there exist a Garside element Δ\Delta in M ?

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  • BRAID GROUPS, CLUSTERS, AND FREE PROBABILITY: AN OUTLINE FROM THE AIM WORKSHOP, JANUARY 2005
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
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    Problem: Reconstructed statement: a cancellative finitely generated monoid MM, with left and right lcms for every pair of elements, must be a classical Garside monoid, i.e. must contain a Garside element Δ\Delta whose left and right divisor sets coincide, are finite, and generate MM. The wording is ambiguous about whether nontrivial units are excluded; the literal text does not exclude them.

    Result: The literal statement is false.

    Let

    M=N×ZM=\mathbb N\times \mathbb Z

    with addition. It is a commutative cancellative monoid, generated by

    (1,0), (0,1), (0,−1).(1,0),\ (0,1),\ (0,-1).

    Divisibility is:

    (a,b)∣(c,d)  ⟺  (c−a,d−b)∈N×Z  ⟺  c≥a.(a,b)\mid(c,d)\iff (c-a,d-b)\in \mathbb N\times\mathbb Z\iff c\ge a.

    Thus any two elements (a,b),(c,d)(a,b),(c,d) have an lcm, unique up to multiplication by a unit, represented by any element with first coordinate max⁡(a,c)\max(a,c). Equivalently,

    ((a,b)+M)∩((c,d)+M)=(max⁡(a,c),0)+M.((a,b)+M)\cap((c,d)+M)=(\max(a,c),0)+M.

    But MM is not a classical Garside monoid. For any Δ=(n,z)\Delta=(n,z), its divisors are

    {(k,t):0≤k≤n, t∈Z},\{(k,t):0\le k\le n,\ t\in\mathbb Z\},

    which is infinite. Hence no element has a finite divisor set, so no classical Garside element exists.

    The defect is the missing “no nontrivial units / conical / atomic-noetherian” hypothesis. Indeed MM has infinite unit group {0}×Z\{0\}\times\mathbb Z.

    A standard repaired theorem is true: if MM is a finitely generated Gaussian monoid—cancellative, conical/atomic, with left and right lcms—then MM is Garside. In that setting the finitely many atoms have a common lcm Δ\Delta, and the Dehornoy–Paris theorem says this Δ\Delta is a Garside element.

    Audit: the counterexample satisfies the literal hypotheses, including finite generation, cancellativity, and existence of lcms; it violates exactly the Garside conclusion.

    Citation: Counterexample elementary. Repaired theorem: P. Dehornoy and L. Paris, “Gaussian groups and Garside groups, two generalisations of Artin groups,” Proc. London Math. Soc. 79 (1999), 569–604.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The counterexample M=N×ZM=\mathbb N\times\mathbb Z is valid for the literal supplied conjecture. It is finitely generated and cancellative, and its principal ideal intersections show that left/right lcms exist up to units. For any Δ=(n,z)\Delta=(n,z), the divisor set contains {(k,t):0≤k≤n, t∈Z}\{(k,t):0\le k\le n,\ t\in\mathbb Z\}, hence is infinite, so no classical Garside element exists. Thus the stated conjecture, without a conical/no-nontrivial-units hypothesis, is false.

    Novelty assessment

    TYPE1

    Classification rationale: The accepted resolution is a valid but very minor counterexample to the literal wording. The issue is simply the omitted “no nontrivial units/conical” hypothesis; indeed even an infinite finitely generated group such as Z\mathbb Z makes the lcm condition degenerate while having no classical Garside element with finite divisor set. This is a routine observation rather than a publishable combinatorial contribution.

    Literature check: I found the standard repaired result in the Garside/Gaussian monoid literature: finitely generated Gaussian-type monoids under the usual noetherian/atomic/conical hypotheses are Garside. I did not find evidence that this exact AIM wording was separately published as “solved” by the N×Z\mathbb N\times\mathbb Z counterexample, but the obstruction from units is standard and immediate from definitions.

    Citation: P. Dehornoy and L. Paris, “Gaussian groups and Garside groups, two generalisations of Artin groups,” Proc. London Math. Soc. 79 (1999), 569–604.

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