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arxiv: 2606.26776 · v2 · pith:YXKDIVGXnew · submitted 2026-06-25 · 🧮 math.QA · math.CO· math.RT

Artin monoids, their homomorphisms and twins

Arkady Berenstein , Jacob Greenstein , Jian-Rong Li This is my paper

Pith reviewed 2026-06-29 05:15 UTC · model grok-4.3

classification 🧮 math.QA math.COmath.RT
keywords Artin monoidsstandard homomorphismstwin homomorphism problemCoxeter groupsHecke monoidsTits conjecturedisjoint homomorphisms
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The pith

Standard homomorphisms of Artin monoids and their compositions yield a large class of solutions to the twin homomorphism problem.

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

The paper constructs many solutions to the twin homomorphism problem for Coxeter groups and their Hecke monoids by starting from standard homomorphisms of Artin monoids. These maps become solutions when composed, provided they are optimal and injective on generators. The construction generalizes the known homogeneous case and the Tits conjecture, which was settled earlier by Crisp and Paris. The authors also classify all pairs of disjoint standard homomorphisms and state a conjecture for the complete list when the source monoid has rank two.

Core claim

We find a large class of its solutions originating from standard homomorphisms of Artin monoids and their compositions. These homomorphisms are expected to be injective when they are optimal and injective on generators, which generalizes the homogeneous homomorphisms and the famous Tits conjecture settled by Crisp and Paris. We classify disjoint standard homomorphisms and conjecture the complete classification when the domain is of rank two.

What carries the argument

standard homomorphisms of Artin monoids that are optimal and injective on generators

If this is right

  • Compositions of such homomorphisms enlarge the known supply of twin solutions beyond the homogeneous case.
  • The Tits conjecture appears as the special case in which the homomorphisms are homogeneous.
  • Disjoint pairs of standard homomorphisms admit an explicit classification.
  • When the source monoid has rank two, the full set of standard homomorphisms is conjecturally classified.

Where Pith is reading between the lines

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

  • The same construction may produce explicit twin solutions for concrete Coxeter groups once the optimal maps are listed.
  • If the rank-two conjecture holds, it would give a complete dictionary of twin maps generated by Artin monoids of low rank.
  • The approach could be tested by checking injectivity for small-rank Artin monoids whose presentations are fully known.

Load-bearing premise

Optimal standard homomorphisms of Artin monoids that are injective on generators must themselves be injective and produce valid twin solutions.

What would settle it

An explicit optimal standard homomorphism between two Artin monoids that is injective on generators yet fails to be injective as a whole or fails to satisfy the twin homomorphism condition.

read the original abstract

Motivated by the twin homomorphism problem for Coxeter groups and the corresponding Hecke monoids, we find a large class of its solutions originating from standard homomorphisms of Artin monoids and their compositions. These homomorphisms are expected to be injective when they are optimal and injective on generators, which generalizes the homogeneous homomorphisms and the famous Tits conjecture settled by Crisp and Paris. We classify disjoint standard homomorphisms and conjecture the complete classification when the domain is of rank two.

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

2 major / 1 minor

Summary. The manuscript claims to find a large class of solutions to the twin homomorphism problem for Coxeter groups and Hecke monoids by constructing them from standard homomorphisms of Artin monoids and their compositions. These homomorphisms are stated to be expected to be injective when optimal and injective on generators, generalizing homogeneous homomorphisms and the Tits conjecture (settled by Crisp and Paris). The paper classifies disjoint standard homomorphisms and conjectures the complete classification in the rank-two case.

Significance. If the expected injectivity can be established, the constructions would supply a broad, explicitly described family of solutions to the twin problem, extending the resolved Tits case. The classification of disjoint standard homomorphisms is a concrete, verifiable contribution to the structure theory of Artin monoid homomorphisms.

major comments (2)
  1. [Abstract] Abstract: the central claim that standard homomorphisms 'supply a large class of its solutions' to the twin problem rests on the assertion that they 'are expected to be injective when they are optimal and injective on generators.' The twin homomorphism problem requires actual injectivity (or an equivalent faithfulness condition) for the maps to be solutions; an unproven expectation does not establish the existence of the claimed class.
  2. [Classification of disjoint standard homomorphisms] The section classifying disjoint standard homomorphisms: while this classification is a solid result, the manuscript only conjectures the complete classification when the domain is of rank two. If the twin-problem solutions are intended to cover the general case, the conjecture must either be proved or the scope of the 'large class' claim must be restricted to the classified cases.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it briefly indicated the criteria used to define 'optimal' homomorphisms and 'disjoint' homomorphisms.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments below and have made revisions to clarify the claims regarding the twin homomorphism problem and the scope of our classification results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that standard homomorphisms 'supply a large class of its solutions' to the twin problem rests on the assertion that they 'are expected to be injective when they are optimal and injective on generators.' The twin homomorphism problem requires actual injectivity (or an equivalent faithfulness condition) for the maps to be solutions; an unproven expectation does not establish the existence of the claimed class.

    Authors: We agree that the twin homomorphism problem requires established injectivity for the homomorphisms to constitute solutions. The manuscript uses 'expected' to indicate that injectivity holds in the known cases (such as the Tits conjecture resolved by Crisp and Paris) and is anticipated to hold more generally under the optimality and generator-injectivity conditions, but we acknowledge that this does not yet prove the existence of the full class of solutions. To address this, we will revise the abstract to describe these as providing a large class of homomorphisms that are expected to yield solutions to the twin problem, rather than claiming they supply solutions outright. This clarifies the status of the injectivity. revision: yes

  2. Referee: [Classification of disjoint standard homomorphisms] The section classifying disjoint standard homomorphisms: while this classification is a solid result, the manuscript only conjectures the complete classification when the domain is of rank two. If the twin-problem solutions are intended to cover the general case, the conjecture must either be proved or the scope of the 'large class' claim must be restricted.

    Authors: The classification of disjoint standard homomorphisms is presented as a complete result in the manuscript for the cases where the homomorphisms are disjoint. The conjecture pertains specifically to the complete classification in the rank-two domain case. To ensure the claims are accurate without requiring a proof of the conjecture at this stage, we will revise the manuscript to restrict the scope of the 'large class' of solutions to those arising from the classified disjoint standard homomorphisms and their compositions, noting that the rank-two case remains conjectural for the full classification. This restricts the general case claim appropriately. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructions and classification are independent of target result

full rationale

The paper constructs standard homomorphisms of Artin monoids, classifies the disjoint ones, and conjectures a full classification for rank-two domains. These are presented as originating solutions to the twin homomorphism problem under an expectation of injectivity that generalizes the externally settled Tits conjecture (Crisp-Paris). No equations, definitions, or self-citations are shown that reduce the claimed class of solutions to a fit, a renaming, or a self-referential premise. The derivation chain remains self-contained against the external benchmark of the Tits result and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities can be identified from the abstract alone; the full manuscript would be required to audit these.

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Reference graph

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