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arxiv: 2604.13989 · v1 · submitted 2026-04-15 · 🧮 math.GR

Computing least common multiples in monoids with a finite Garside family

Emir Melliti This is my paper

Pith reviewed 2026-05-10 11:58 UTC · model grok-4.3

classification 🧮 math.GR
keywords monoidsGarside familiesright-reversingleast common multiplescyclic runsright-complemented presentationsalgorithm termination
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The pith

A non-terminating right-reversing run in a monoid is necessarily cyclic, allowing computation of a minimal Garside family by detecting the cycle.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Right-reversing is an algorithm for computing least common multiples in monoids that have right-complemented presentations. The paper proves that if this algorithm runs indefinitely, its execution must eventually cycle. Stopping the run upon detecting the cycle then produces a minimal Garside family. Under the Garside theory assumptions, this extends the algorithm's utility to cases where it would otherwise not terminate.

Core claim

In the framework of monoids admitting right-complemented presentations with Garside assumptions, we establish that any non-terminating execution of the right-reversing algorithm must be cyclic. Detecting such a cycle and stopping the run thereby yields a minimal Garside family for the monoid.

What carries the argument

The right-reversing algorithm applied to a right-complemented presentation, with cycle detection used to extract the minimal Garside family.

If this is right

  • The algorithm terminates with a minimal Garside family when a cycle is detected in an otherwise infinite run.
  • Least common multiples can be computed using the resulting Garside family.
  • Monoids satisfying the right-complemented and Garside conditions now have a method to find their minimal Garside family algorithmically.
  • The proof relies on the same framework that establishes correctness of terminating runs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Implementing cycle detection in the algorithm provides a practical decision procedure for whether a given monoid admits a finite Garside family.
  • This approach might generalize to other non-terminating rewriting procedures in algebra by identifying repeating states.
  • Examples like braid monoids could be tested to see if the method recovers known Garside families efficiently.

Load-bearing premise

The monoid admits a right-complemented presentation and satisfies the additional Garside theory assumptions required for the algorithm's correctness.

What would settle it

Finding a monoid with a right-complemented presentation where the right-reversing algorithm runs indefinitely without any repetition of a state, despite satisfying the Garside conditions.

Figures

Figures reproduced from arXiv: 2604.13989 by Emir Melliti.

Figure 1
Figure 1. Figure 1: Illustration of the proof of Lemma 2.9. Proof of Proposition 2.8. Let ad = bc be a common right-multiple of a and b. Put e = c ∧e d, write c = c ′ e and d = d ′ e, so that c ′ ∧e d ′ = 1. Then by Lemma 2.9, ac′ = bd′ is a right-lcm of a and b [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the proof of Lemma 2.10. See the similarity with the concatenation of pushout [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the proof of Lemma 3.9 (3). [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Graph obtained by applying the reduction rule to the initial assumption [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Completing the graph from figure 4 to get a Van Kampen diagram. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Graph obtained by applying the reduction rule to the initial assumption [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of the right-reversing process. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: On the left, applying the reduction rule, and on the right, applying right-reversing, to [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Illustration of the notion of quadruple factorable through right-reversing. [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The θ-cube condition. When trying to prove the reduction rule, (or the completeness of right-reversing), the proof works on a nested induction, first on the length of the common multi￾ple aX = bY , and then on the length of a shortest derivation from aX to bY . In the induction step appears the relation aX . cZ . bY , and applying the reduc￾tion rule to aX . cZ and cZ . bY gives two expressions for Z, θ(c… view at source ↗
Figure 11
Figure 11. Figure 11: The right-reversing diagram witnessing the non-termination of the right-reversing process [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Illustration of the proof of Lemma 4.18. [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Illustration of the proof of Lemma 4.19. [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The smallest Garside family for the affine type Artin-Tits monoid of type [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The smallest Garside family for the affine type Artin-Tits monoid of type [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The smallest Garside family for the affine type Artin-Tits monoid of type [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
read the original abstract

Right-reversing is an algorithm used to compute least common multiples in monoids that admit a right-complemented presentation. The algorithm can either terminate and find a result, fail, or run indefinitely. The correctness of the algorithm can be proved with additional assumptions coming from Garside theory. In the same framework, we prove that a non-terminating run of the algorithm is necessarily cyclic. Stopping when a cycle is detected provides a way of computing a minimal Garside family.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops the right-reversing algorithm for computing least common multiples in monoids that admit a right-complemented presentation. Under the standard Garside-theoretic hypotheses (including the existence of a finite Garside family), it proves that any non-terminating run of the algorithm must eventually become cyclic. It then shows that detecting such a cycle yields a method for computing a minimal Garside family.

Significance. If the central proof is correct, the result supplies a concrete algorithmic procedure for obtaining minimal Garside families in a class of monoids where they are known to exist but may be difficult to exhibit by hand. This strengthens the computational toolkit available within Garside theory and may facilitate systematic study of monoids and groups arising from right-complemented presentations.

minor comments (3)
  1. §2.3: the statement of the right-complemented property is given only by reference to an earlier paper; a self-contained one-sentence reminder of the exact definition used here would improve readability for readers who are not specialists in the area.
  2. Definition 3.4 and the surrounding discussion: the notation for the reversing sequence (e.g., the distinction between the partial products and the full word) is introduced without an explicit running example; adding a short worked example of a non-terminating sequence would clarify the subsequent periodicity argument.
  3. Theorem 4.1: the proof sketch invokes the finiteness of the Garside family to bound the set of divisors, but does not explicitly record the resulting bound on the length of the transient before the cycle appears; stating this bound would make the termination argument for the cycle-detection procedure fully explicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for recommending minor revision. The report provides a concise summary of the main results but does not list any specific major comments or points requiring clarification.

Circularity Check

0 steps flagged

No significant circularity detected in the proof

full rationale

The paper presents a deductive proof within Garside theory that non-terminating right-reversing runs on monoids with right-complemented presentations must be cyclic, enabling computation of minimal Garside families via cycle detection. This relies on standard finiteness and complementation hypotheses from the established framework rather than any self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation chain is self-contained and does not reduce any claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger records the two domain assumptions explicitly stated there; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Monoids admit a right-complemented presentation
    Required for the right-reversing algorithm to be defined and applicable.
  • domain assumption Additional Garside-theory assumptions hold
    Invoked to guarantee correctness of the algorithm and the cycle property.

pith-pipeline@v0.9.0 · 5362 in / 1106 out tokens · 38161 ms · 2026-05-10T11:58:58.526947+00:00 · methodology

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