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arxiv: 2507.04819 · v2 · submitted 2025-07-07 · 🧮 math.GR · math.RA

Two-sided homological properties of special and one-relator monoids

Robert D. Gray , Benjamin Steinberg This is my paper

Pith reviewed 2026-05-19 06:42 UTC · model grok-4.3

classification 🧮 math.GR math.RA
keywords special monoidsone-relator monoidshomological finiteness propertiesHochschild cohomological dimensiongroup of unitsbi-FP_ntwo-sided homology
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The pith

Special monoids have bi-FP_n if their groups of units do.

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

The paper proves that monoids defined by special presentations, where each relation ends with 1, have their two-sided homological finiteness properties determined by those of the group of units. If the group of units is of type FP_n, then the monoid is bi-FP_n. The Hochschild cohomological dimension of the monoid is at most the maximum of 2 and the cohomological dimension of its group of units. Applying this to one-relator monoids of the form , the paper shows they are all bi-FP_infty, with the dimension being at most 2 if r is not a proper power and infinite otherwise. For non-special one-relator monoids without a common overlapping word factor, similar finiteness and dimension bounds hold.

Core claim

Monoids with special presentations enjoy the homological finiteness property bi-FP_n whenever their group of units is of type FP_n. The Hochschild cohomological dimension is bounded above by max(2, the cohomological dimension of the group of units). All one-relator monoids <A | r=1> are of type bi-FP_infty. If the relator r is not a proper power, the Hochschild dimension is at most 2; if it is a proper power, the dimension is infinite. Non-special one-relator monoids without the word overlap condition are bi-FP_infty with dimension at most 2.

What carries the argument

The special presentation, where the right-hand side of each relation is 1, which permits relating the monoid's two-sided resolutions to the resolutions of its group of units.

If this is right

  • If the group of units is FP_n then the monoid is bi-FP_n.
  • The Hochschild cohomological dimension of the monoid is at most max(2, cd of units group).
  • One-relator monoids <A|r=1> are bi-FP_infty.
  • If r not proper power, Hochschild cd <=2; if proper power, cd infinite.
  • Non-special one-relator monoids without overlap word are bi-FP_infty with cd <=2.

Where Pith is reading between the lines

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

  • The reduction from monoid to group homological properties may apply to broader classes of monoids with restricted presentations.
  • This could lead to new ways to compute homology for monoids by first finding the unit group.
  • Concrete examples of one-relator monoids can be used to test the dimension bounds.

Load-bearing premise

The monoid must admit a special presentation or be a one-relator monoid satisfying the no common factor overlap condition.

What would settle it

Find a monoid with special presentation where the group of units has type FP_n but the monoid fails to be bi-FP_n, or has Hochschild dimension exceeding max(2, cd of the group).

Figures

Figures reproduced from arXiv: 2507.04819 by Benjamin Steinberg, Robert D. Gray.

Figure 1
Figure 1. Figure 1: Homological finiteness properties satisfied by monoids that admit finite complete rewriting systems, and all implications between them. The proofs that all these implications hold, and that none of the implications in the diagram are reversible, can be found in the papers [14, 34, 36, 38, 39, 53, 54, 58]. spaces for one-relator groups that can be used to deduce all one-relator groups are FP8, and to determ… view at source ↗
read the original abstract

A monoid presentation is called special if the right-hand side of each defining relation is equal to 1. We prove results which relate the two-sided homological finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid enjoys the homological finiteness property bi-$\mathrm{FP}_n$ if its group of units is of type $\mathrm{FP}_n$. We also obtain results which relate the Hochschild cohomological dimension of the monoid to the cohomological dimension of its group of units. In particular we show that the Hochschild cohomological dimension of the monoid is bounded above by the maximum of 2 and the cohomological dimension of its group of units. We apply these results to prove a Lyndon's Identity type theorem for the two-sided homology of one-relator monoids of the form $\langle A \mid r=1 \rangle$. In particular, we show that all such monoids are of type bi-$\mathrm{FP}_\infty$. Moreover, we show that if $r$ is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most $2$, while if $r$ is a proper power then it has infinite Hochschild cohomological dimension. For any non-special one-relator monoid $M$ with defining relation $u=v$ we show that if there is no nonempty word $w$ such that $u,v \in A^*w \cap w A^*$ then $M$ is of type bi-$\mathrm{FP}_\infty$ and has Hochschild cohomological dimension at most $2$.

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 paper proves that monoids with special presentations (relations of the form r=1) satisfy the two-sided homological finiteness property bi-FP_n whenever their group of units is of type FP_n. It further shows that the Hochschild cohomological dimension of such a monoid is at most max(2, cd(G)), where G is the group of units. These transfer results are applied to one-relator monoids <A | r=1>, establishing that all such monoids are of type bi-FP_∞, with Hochschild dimension ≤2 if r is not a proper power and infinite otherwise. Analogous bi-FP_∞ and dimension ≤2 results are obtained for non-special one-relator monoids <A | u=v> under the assumption that u and v share no common nonempty prefix-suffix overlap word w.

Significance. If the central claims hold, the work supplies a concrete mechanism for lifting homological finiteness and dimension bounds from groups to monoids within the restricted classes of special and overlap-controlled one-relator presentations. The explicit construction relating the two-sided bar resolution (or a custom bimodule resolution) of the monoid to the corresponding resolution of its group of units is a clear technical strength, as is the resulting Lyndon-type theorem for one-relator monoids and the sharp distinction on proper powers. These results are likely to be useful for classifying monoids with controlled two-sided homology.

minor comments (3)
  1. The abstract and introduction state the overlap condition for non-special one-relator monoids clearly, but the precise formulation of the bimodule resolution used to prove the bi-FP_n transfer (likely in the section following the statement of the main theorems) would benefit from an explicit diagram or short exact sequence that isolates the contribution of the group-of-units resolution.
  2. Notation for the two-sided bar resolution and the Hochschild cochain complex is introduced without a dedicated preliminary subsection; adding a short paragraph recalling the standard definitions (with references to standard texts) would improve readability for readers outside homological algebra.
  3. The proof that the Hochschild dimension is infinite when r is a proper power relies on exhibiting a non-vanishing cohomology class; a brief indication of the explicit cocycle or the spectral-sequence argument used would make the infinite-dimensionality claim easier to verify at a glance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive evaluation of its significance. The referee's summary accurately reflects the main theorems on bi-FP_n transfer for special monoids and the Lyndon-type results for one-relator monoids. We are pleased with the recommendation for minor revision and note that no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations use standard homological constructions

full rationale

The paper derives bi-FP_n transfer and Hochschild dimension bounds for special monoids and one-relator monoids by relating their two-sided resolutions to those of the group of units via explicit bimodule constructions and overlap conditions. These steps rely on standard bar resolutions and Lyndon-type arguments rather than self-definitional equivalences, fitted parameters renamed as predictions, or load-bearing self-citations that presuppose the target results. The one-relator applications (bi-FP_infty, dimension distinctions for proper powers, and non-special overlap cases) follow directly as corollaries without circular reduction to inputs. The derivation chain remains self-contained against external homological algebra benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard homological algebra for monoids and groups without introducing new free parameters, invented entities, or ad-hoc axioms beyond established background results in the field.

axioms (1)
  • standard math Standard results from homological algebra for monoids and their groups of units hold, including existence of resolutions and properties of Hochschild cohomology.
    Invoked throughout to relate monoid and group resolutions.

pith-pipeline@v0.9.0 · 5838 in / 1302 out tokens · 47025 ms · 2026-05-19T06:42:18.109705+00:00 · methodology

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