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arxiv: 2606.21106 · v1 · pith:CY6PBIASnew · submitted 2026-06-19 · 🧮 math.GR

The word problem for a family of one relation Adian inverse semigroups

Muhammad Inam This is my paper

Pith reviewed 2026-06-26 13:00 UTC · model grok-4.3

classification 🧮 math.GR
keywords word probleminverse semigroupsAdian semigroupsone-relation presentationsdecidabilitysemigroup presentations
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The pith

The word problem is decidable for Adian inverse semigroups presented by Inv⟨a,b|a=ba^nb⟩ for any n≥1.

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

The paper establishes decidability of the word problem in the inverse semigroup defined by the one-relation presentation Inv⟨a,b | a = b a^n b⟩. This means an algorithm exists that can determine whether any two words over the generators a and b represent the same element. A sympathetic reader would care because the word problem is undecidable in many algebraic structures, and explicit decidable families help delineate the boundary between solvable and unsolvable cases.

Core claim

The word problem for an Adian inverse semigroup given by the presentation Inv⟨a,b|a=ba^nb⟩, where n≥1, is decidable.

What carries the argument

The one-relation presentation Inv⟨a,b|a=ba^nb⟩ that defines the Adian inverse semigroup and supports construction of a decision procedure for word equality.

If this is right

  • An algorithm exists that decides equality of any two words in these semigroups.
  • The semigroup elements can be represented and compared by a finite procedure.
  • The relation a = b a^n b admits effective reduction rules for words.

Where Pith is reading between the lines

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

  • Similar techniques could apply to other one-relation inverse semigroup presentations.
  • The result supplies a concrete infinite family of inverse semigroups with solvable word problems.
  • It may support computational checks of semigroup identities for these presentations.

Load-bearing premise

The specific one-relation presentation Inv⟨a,b|a=ba^nb⟩ defines an Adian inverse semigroup whose structure permits an effective decision procedure for equality of words.

What would settle it

An explicit pair of words over a and b, for some fixed n, such that no algorithm can decide whether they represent the same element in the semigroup.

Figures

Figures reproduced from arXiv: 2606.21106 by Muhammad Inam.

Figure 1
Figure 1. Figure 1: displays the pivotal transversal, the edges of color 0 and color 1 in MT(w1), for some w1 ∈ (X ∪ X−1 ) ∗ , with respect to the presentation hX|a = ba3 bi. a a a b b Edges of Color 0 Edge of Color 1 Pivotal Transversal b [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: MT(w2), for some w2 ∈ (X ∪ X−1 ) ∗ Proposition 2. An edge of color 0 of MT(w) that contains only one pivotal transversal, for some w ∈ (X ∪ X−1 ) ∗ , is not a boundary edge of any region of SΓ(w). Proof. The proof consists of two parts. The first part deals with those max￾imal segments of the form b r of MT(w) that satisfy all the three conditions given in the definition of edges of color 0. The second par… view at source ↗
Figure 3
Figure 3. Figure 3: MT(w1) with three pivotal transversals and p = 2, for some w ∈ (X ∪ X−1 ) ∗ It is possible that some pivotal transversals may not contain any cut vertex, while some other pivotal transversals may contain more than one cut vertex. We label these cut vertices of MT(w) by γ1, γ2, ..., γ(q−1) (see figure 3). We distribute MT(w) into q fragments by cutting MT(w) at its cut vertices. Each of these fragments cont… view at source ↗
Figure 4
Figure 4. Figure 4: MT(w) distributed into three fragments We assume that p1, p2, ..., pq be the maximums of the lengths of the di￾rected segments of the form b r , oriented in any direction, of the fragments Π1, Π2, ..., Πq , respectively. We apply full P-expansions on each Πi succes￾sively up to (pi+ 1)-times, and denote the corresponding resulting graphs by Λ1,Λ2, ...,Λq , respectively (see figure 5). By Lemma 1, Λi ’s do … view at source ↗
Figure 5
Figure 5. Figure 5: Λi obtained from Πi , for 1 ≤ i ≤ 3, over the presentation hX|a = babi b b b b b b b b b b b b b b b b a a a a a a a a a a a a a a a a b b b b b b b b Δ1 Δ2 Δ b b [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: ∆i obtained from Λi , for 1 ≤ i ≤ 3, over the presentation hX|a = babi By Lemma 3, none of the finite graphs ∆1, ∆2, ..., ∆q contain any unsat￾urated special vertex relative to the corresponding Πi , where 1 ≤ i ≤ q. We connect these finite graphs with each other by connecting each γi , for 1 ≤ i ≤ q − 1, at the same vertex of the pivotal transversal of ∆j , for 1 ≤ j ≤ q, from where it was removed. We den… view at source ↗
Figure 7
Figure 7. Figure 7: Θ obtained by connecting all ∆j ’s at γi ’s and performing all possible foldings special vertices, then such a segment will not contain any unsaturated special vertex in Θ. Therefore, we can ignore all such connecting segments. It is easy to see that if a maximal connecting segment (that is not a proper subsegment of any larger segment of the form b r in MT(w)) is con￾necting more than two pivotal transver… view at source ↗
Figure 8
Figure 8. Figure 8: Γ constrcted from Θ by using the above iterative process, over the presentation hX|a = babi special vertex. By Lemma 3, it follows that for j ≥ m, none of the approx￾imate graph (αj , Γj (w), βj ) contains any unsaturated special vertex. So, (αi , Γi(w), βi) embeds in (αi+1, Γi+1(w1), βi+1) for i ≥ m, by Lemma 2. This leads to the conclusion that for any vertex γ ∈ V ((αm+i , Γm+i(w), βm+i) \ (αm+i−1, Γm+i… view at source ↗
read the original abstract

The word problem for an Adian inverse semigroup given by the presentation $Inv\langle a,b|a=ba^nb\rangle$, where $n\geq 1$, is decidable.

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

1 major / 0 minor

Summary. The manuscript asserts that the word problem for the Adian inverse semigroup presented by Inv⟨a,b | a=ba^nb⟩ (n≥1) is decidable.

Significance. A positive result on decidability for this specific one-relation family would be of interest in inverse semigroup theory, where word problems are frequently undecidable, but the absence of any supporting argument limits assessment of its potential impact.

major comments (1)
  1. The manuscript text consists solely of a one-sentence abstract stating the decidability claim, with no proof sketch, outline of the decision procedure, structural analysis of the semigroup, or reference to any construction or algorithm. This prevents verification of the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We agree that the submitted manuscript contains only the statement of the result and lacks supporting arguments, and we will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The manuscript text consists solely of a one-sentence abstract stating the decidability claim, with no proof sketch, outline of the decision procedure, structural analysis of the semigroup, or reference to any construction or algorithm. This prevents verification of the central claim.

    Authors: We agree with the referee that the current manuscript provides only the claim without any proof, algorithm, or analysis. This was an error in the submission. The revised manuscript will include the full proof of decidability, the decision procedure, and the necessary structural analysis of the semigroup. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript states a decidability result for the word problem in the specific one-relation Adian inverse semigroup Inv⟨a,b|a=ba^nb⟩ (n≥1). No equations, parameter fits, self-citations, or ansatzes are exhibited in the abstract or described structure that reduce the claimed decision procedure to its own inputs by construction. The derivation is therefore treated as self-contained and externally verifiable via explicit construction of the algorithm.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on domain assumptions from inverse semigroup theory that the given presentation yields an Adian inverse semigroup; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption The presentation Inv⟨a,b|a=ba^nb⟩ defines an Adian inverse semigroup with the standard structural properties used in word-problem studies.
    The claim presupposes that the given relation produces an object to which known or new decision methods apply.

pith-pipeline@v0.9.1-grok · 5534 in / 1126 out tokens · 44763 ms · 2026-06-26T13:00:49.180558+00:00 · methodology

discussion (0)

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Reference graph

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