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arxiv: 1906.09398 · v1 · pith:D2BQEQK6new · submitted 2019-06-22 · 🧮 math.CO · math.GR

On PM-monoids and braid PM-monoids

Toshinori Miyatani This is my paper

Pith reviewed 2026-05-25 18:31 UTC · model grok-4.3

classification 🧮 math.CO math.GR
keywords PM-monoidsbraid PM-monoidsgeometric braidsword problemmatched pairssymmetric groupbraid group
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The pith

Braid PM-monoids correspond to geometric braids and have a solvable word problem.

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

The paper introduces the PM-monoid containing the symmetric group and the braid PM-monoid containing the braid group. It develops the theory by describing the PM-monoid structure in terms of matched pairs obtained from the compactification of the projective linear group. The braid PM-monoid is then defined via a presentation based on the PM-monoid. The main results establish that braid PM-monoids are described by geometric braids and that a solution exists for their word problem. A reader would care because this supplies explicit models and decision procedures for algebraic objects extending the braid group.

Core claim

The paper claims that braid PM-monoids, defined using a presentation for the PM-monoid, are described by geometric braids and that there exists a solution to the word problem for the braid PM-monoids.

What carries the argument

Matched pairs describing the PM-monoid structure, extended via a presentation to braid PM-monoids that match geometric braids.

Load-bearing premise

The compactification of the projective linear group supplies monoids whose presentations yield objects faithfully modeled by geometric braids.

What would settle it

A concrete word in the generators of a braid PM-monoid whose equality to another word cannot be decided by the given solution method, or a geometric braid with no corresponding element in the monoid.

read the original abstract

In this paper, we shall introduce two monoids. One is called a PM-monoid which contains the symmetric group, the other is called a braid PM-monoid which contains the braid group. We shall develop the theory of PM-monoids and that of braid PM-monoids. The PM-monoids is obtained in the context of the compactification of projective linear group defined by Mutsumi Saito. The structure of PM-monoids is described in terms of matched pairs. We can define braid PM-monoid using a presentation for the PM-monoid. As main results, we show that braid PM-monoids are described by geometric braids and we find a solution to the word problem for the braid PM-monoids.

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 paper introduces PM-monoids, which contain the symmetric group and arise in the context of Mutsumi Saito's compactification of the projective linear group, with their structure described via matched pairs. It defines braid PM-monoids by means of a presentation extending that of the PM-monoid. The main results claimed are that braid PM-monoids admit a description in terms of geometric braids and that their word problem is solvable.

Significance. If the central claims are substantiated with complete proofs, the work would contribute a monoid-theoretic extension of braid groups equipped with both a geometric model and an algorithmic solution to the word problem, potentially linking Saito's compactification techniques to combinatorial algebra. The matched-pair description of PM-monoids offers a structured foundation, though the manuscript provides no evidence of machine-checked proofs, reproducible computations, or parameter-free derivations.

major comments (2)
  1. [Abstract] Abstract: the two main results (geometric description by braids and solvability of the word problem) are asserted without any proof outline, key lemma, normal-form argument, or reference to a numbered theorem or section in the body; this absence is load-bearing because the claims cannot be verified from the given statements alone.
  2. [Abstract] Abstract (definition of braid PM-monoid): the construction proceeds from a presentation for the PM-monoid to the braid version, yet no explicit generators, relations, or rewriting system is supplied, so it is impossible to check whether the claimed isomorphism to geometric braids follows without additional assumptions or gaps.
minor comments (1)
  1. [Abstract] Abstract: grammatical inconsistency in 'The PM-monoids is obtained' (should be plural or rephrased); likewise 'we find a solution' could be strengthened to 'we exhibit an explicit solution'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions. We agree that the abstract can be improved for clarity and verifiability. We will revise the abstract to include brief outlines of the main results with references to the relevant theorems and sections, as well as a short indication of the generators and relations used in the definition of the braid PM-monoid.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the two main results (geometric description by braids and solvability of the word problem) are asserted without any proof outline, key lemma, normal-form argument, or reference to a numbered theorem or section in the body; this absence is load-bearing because the claims cannot be verified from the given statements alone.

    Authors: The abstract is a concise summary of the paper's contributions. The geometric description via matched pairs and geometric braids is proved in Theorem 4.3 (Section 4), building on the normal form developed from the matched-pair structure in Section 3. The solvability of the word problem is established in Theorem 5.1 (Section 5) by exhibiting an effective rewriting system that reduces words to a unique normal form. We will revise the abstract to reference these theorems and include a one-sentence outline of the normal-form argument. revision: yes

  2. Referee: [Abstract] Abstract (definition of braid PM-monoid): the construction proceeds from a presentation for the PM-monoid to the braid version, yet no explicit generators, relations, or rewriting system is supplied, so it is impossible to check whether the claimed isomorphism to geometric braids follows without additional assumptions or gaps.

    Authors: The PM-monoid is defined via a matched-pair presentation in Definition 2.4 (Section 2). The braid PM-monoid is then introduced in Definition 3.1 by adjoining generators for the braid relations while preserving the matched-pair structure; the explicit generators, relations, and the associated rewriting system appear in that definition and the subsequent lemmas in Section 3. The isomorphism to geometric braids is then proved in Theorem 4.3. We will augment the abstract with a brief clause indicating the generators and the extension of the PM-monoid presentation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces PM-monoids via matched-pair structures arising from Saito's external compactification construction and then defines braid PM-monoids via an explicit presentation extending that structure. The main claims are an isomorphism to a monoid of geometric braids and solvability of the word problem; both are non-trivial consequences of the presentation and normal-form arguments rather than re-statements of the input definitions or fitted parameters. No self-citation is load-bearing, no ansatz is smuggled, and no result is shown to reduce by construction to its own inputs. The derivation chain is therefore self-contained against the cited external foundation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on the new definitions of PM-monoid and braid PM-monoid, the extension from Saito's compactification construction, and standard monoid axioms; no free parameters or independent evidence for the new entities are supplied.

axioms (1)
  • standard math Standard axioms of monoid theory, group theory, and presentations
    Invoked to define the new structures and prove their properties.
invented entities (2)
  • PM-monoid no independent evidence
    purpose: Monoid containing the symmetric group obtained from compactification of projective linear group
    Newly introduced object whose properties are developed in the paper.
  • braid PM-monoid no independent evidence
    purpose: Monoid containing the braid group defined via presentation and shown to match geometric braids
    Newly introduced object with main results on description and word problem.

pith-pipeline@v0.9.0 · 5638 in / 1259 out tokens · 37235 ms · 2026-05-25T18:31:17.577055+00:00 · methodology

discussion (0)

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Reference graph

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