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arxiv: 2301.04252 · v3 · submitted 2023-01-11 · 🧮 math.GR

Conjugacy in Abstract Semigroups, Transformation and Diagram Monoids, and Conjugacy Growth

Jo\~ao Ara\'ujo , Wolfram Bentz , Michael Kinyon , Janusz Konieczny , Ant\'onio Malheiro , Valentin Mercier This is my paper

Pith reviewed 2026-05-24 10:39 UTC · model grok-4.3

classification 🧮 math.GR
keywords conjugacysemigroupstransformation monoidsdiagram monoidspolycyclic monoidsclass classificationgrowth function
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The pith

The four-equation conjugacy relation cf n admits complete class classifications in the full transformation monoid T_n and the symmetric inverse monoid I_n.

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

The paper defines a conjugacy relation cf n on semigroups by the existence of elements g and h satisfying four specific equations. It analyzes how this relation interacts with other conjugacy notions in abstract semigroups before specializing to concrete families. Complete classifications of the resulting equivalence classes are given for the full transformation monoid, the symmetric inverse monoid, and endomorphism monoids of G-sets. The same relation is applied to diagram semigroups including partition, Brauer, and partial Brauer monoids, while a precise asymptotic is derived for the conjugacy growth function on polycyclic monoids.

Core claim

The relation a cf n b holds when there exist g, h in S^1 such that ag = gb, bh = ha, hag = b, and gbh = a; this yields exhaustive enumerations of equivalence classes in T_n, I_n, and related monoids, together with structural results on natural conjugacy in diagram semigroups and an exact growth estimate in polycyclic monoids.

What carries the argument

The cf n relation, the four-equation definition that generates equivalence classes whose structure is classified by direct combinatorial arguments in the listed monoids.

If this is right

  • Every cf n-class in T_n can be listed by rank and image type.
  • The same enumeration applies without change to I_n and to endomorphism monoids of finite G-sets.
  • Natural conjugacy on partition monoids, Brauer monoids and partial Brauer monoids is completely determined by the same four equations.
  • The conjugacy growth function of any polycyclic monoid admits an exact asymptotic formula.

Where Pith is reading between the lines

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

  • The four-equation definition may serve as a uniform test for conjugacy across other families of finite monoids not treated in the paper.
  • The classification techniques could be adapted to decide cf n-membership algorithmically for monoids given by presentations.
  • Growth estimates obtained for polycyclic monoids suggest that similar asymptotics might exist for related monoids with solvable word problems.
  • The interplay results with standard conjugacy relations indicate that cf n could refine existing invariants used in automata theory.

Load-bearing premise

The four equations produce equivalence classes whose complete listing in T_n, I_n and the diagram monoids requires no additional hidden constraints or missed case distinctions.

What would settle it

An explicit pair of elements in T_3 whose membership in the same cf n-class contradicts the listed classification tables, or a pair that satisfies cf n but violates one of the four defining equations.

Figures

Figures reproduced from arXiv: 2301.04252 by Ant\'onio Malheiro, Janusz Konieczny, Jo\~ao Ara\'ujo, Michael Kinyon, Valentin Mercier, Wolfram Bentz.

Figure 4
Figure 4. Figure 4: presents an example of a functional digraph, its trim, and [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: A functional digraph (left), its trim (middle), and its prun [PITH_FULL_IMAGE:figures/full_fig_p018_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Γ(α) (left), Γ(β) (middle), and Γ(δ) (right). • • [PITH_FULL_IMAGE:figures/full_fig_p021_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Γp (α) (left), Γp (β) (middle), and Γp (δ) (right). 21 [PITH_FULL_IMAGE:figures/full_fig_p021_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Γp β (α) (left) and Γp δ (α) (right). Recall that for an integer n ≥ 1, Xn = {1 < . . . < n}. Viewing Xn as a set, we denote by In the symmetric inverse semigroup I(Xn). Let OIn be the subset of In consisting of partial injective order￾preserving transformations, that is, OIn = {α ∈ In : ∀x,y∈dom(α)(x < y ⇒ xα < yα)}. Then OIn is an inverse semigroup [53, 54]. We will now describe n-conjugacy in OIn. Let… view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: The directed graph of an idempotent. Let ρ be an equivalence relation on X, and let R be a cross-section of the partition X/ρ induced by ρ. Then, the set T (X, ρ, R) of elements α ∈ T (X) that preserve both ρ and R, T (X, ρ, R) = {α ∈ T (X) : Rα ⊆ R and (x, y) ∈ ρ =⇒ (xα, yα) ∈ ρ}, is a subsemigroup of T (X). The semigroups T (X, ρ, R) are exactly the same as the centralizers C(ε) of idempotents ε ∈ T (X… view at source ↗
read the original abstract

We study conjugacy relations on semigroups and monoids, focusing on the relation $a \cfn b$, defined by the existence of $g,h \in S^1$ such that $ag = gb$, $bh = ha$, $hag = b$, and $gbh = a$. This notion emerged as one that yields particularly elegant results. The interplay between $\cfn$ and other standard conjugacy relations is analyzed, and some results on special classes of abstract semigroups are established. We then specialize to the case of transformation semigroups. A complete classification of $\cfn$-classes is obtained for the full transformation monoid $\mathcal{T}_n$, the symmetric inverse monoid $\mathcal{I}_n$, and the endomorphism monoid of $G$-sets, among others. We also investigate the natural conjugacy in diagram semigroups, including the partition monoid, the Brauer monoid, and the partial Brauer monoid. Finally, we investigate the conjugacy growth function in polycyclic monoids and obtain a precise asymptotic estimate. The paper concludes with some open problems.

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 / 2 minor

Summary. The paper introduces the conjugacy relation a cf_n b on a semigroup S, defined by the existence of g, h in S^1 satisfying the four equations ag = gb, bh = ha, hag = b, and gbh = a. It analyzes the relation's interplay with other conjugacy notions in abstract semigroups, then specializes to obtain a complete classification of cf_n-classes in the full transformation monoid T_n, the symmetric inverse monoid I_n, and the endomorphism monoid of G-sets. The work further studies natural conjugacy in diagram semigroups (partition, Brauer, and partial Brauer monoids) and derives a precise asymptotic estimate for the conjugacy growth function in polycyclic monoids, concluding with open problems.

Significance. If the classifications are exhaustive, the paper would advance semigroup theory by exhibiting a conjugacy relation that produces clean structural results in key monoids and by supplying explicit descriptions and growth asymptotics that can serve as benchmarks for further work on conjugacy invariants.

major comments (2)
  1. [Section on transformation semigroups (classification for T_n and I_n)] The central claim of a complete classification of cf_n-classes in T_n (and similarly I_n) rests on exhaustive enumeration via the four-equation definition; the case analysis by rank, image, and kernel must be shown to cover every configuration without omitted cases or additional constraints imposed by the equations hag = b and gbh = a that would alter the partition into classes.
  2. [Section on diagram semigroups] For the diagram monoids, the investigation of natural conjugacy should explicitly verify that the four equations reduce to the expected combinatorial conditions on diagrams without introducing extra equivalences not captured by the standard diagrammatic description.
minor comments (2)
  1. [Introduction] The abstract states that cf_n 'emerged as one that yields particularly elegant results'; a brief comparison table or explicit statement of which prior relations (e.g., ~_L, ~_R, ~_J) it refines would help readers situate the new relation.
  2. [Section on polycyclic monoids] Notation for the polycyclic monoids and the precise statement of the asymptotic estimate for the conjugacy growth function should be cross-referenced to the relevant theorem number for quick lookup.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comments point by point below, and we will revise the manuscript to incorporate explicit verifications as suggested.

read point-by-point responses

  1. Referee: [Section on transformation semigroups (classification for T_n and I_n)] The central claim of a complete classification of cf_n-classes in T_n (and similarly I_n) rests on exhaustive enumeration via the four-equation definition; the case analysis by rank, image, and kernel must be shown to cover every configuration without omitted cases or additional constraints imposed by the equations hag = b and gbh = a that would alter the partition into classes.

    Authors: The manuscript presents a complete classification through exhaustive case analysis on the rank, image, and kernel of the transformations in T_n and I_n. We have ensured that the four equations are fully accounted for in determining the equivalence classes. To address the referee's concern and make the exhaustiveness explicit, we will add a new lemma that systematically verifies that all possible configurations are covered and that the equations hag = b and gbh = a do not impose additional constraints beyond those used in the classification. This revision will be included in the updated version of the paper. revision: yes

  2. Referee: [Section on diagram semigroups] For the diagram monoids, the investigation of natural conjugacy should explicitly verify that the four equations reduce to the expected combinatorial conditions on diagrams without introducing extra equivalences not captured by the standard diagrammatic description.

    Authors: In the section on diagram semigroups, the natural conjugacy is defined and analyzed by reducing the four equations to conditions on the partitions and connections represented by the diagrams. We believe this reduction is accurate and does not introduce extra equivalences. However, to provide the explicit verification requested, we will add a short proposition in the revised manuscript that derives the combinatorial conditions directly from the equations, confirming they match the expected description without additional relations. revision: yes

Circularity Check

0 steps flagged

No circularity: direct classification from explicit four-equation definition

full rationale

The paper introduces the cf_n relation via an explicit four-equation definition (ag=gb, bh=ha, hag=b, gbh=a) and proceeds by direct case analysis on the combinatorial structure of the target monoids (T_n, I_n, diagram monoids, etc.). No parameters are fitted to data, no predictions are renamed from inputs, and no load-bearing steps reduce to self-citations or prior ansatzes by the same authors. The completeness claim rests on exhaustive enumeration within the given relation, which is an independent mathematical argument rather than a definitional tautology. This is the normal case of a self-contained classification theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard definition of semigroups and monoids together with the explicit four-equation definition of the cf n relation; no free parameters, invented entities, or non-standard axioms are indicated.

axioms (1)
  • standard math A semigroup is a set equipped with an associative binary operation.
    Invoked throughout the study of abstract semigroups and their specializations.

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Reference graph

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