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In Search of Homology for Quasigroups of Bol-Moufang Type

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Pith's one-line read This paper constructs the first homology groups for quasigroups of Bol-Moufang type, defining boundary maps from extensions by affine quasigroups and computing H1 and H2 for the examples that distinguish all 26 varieties.

desk verdict New boundary maps give the first H1/H2 for Bol-Moufang quasigroups, but the key variety-independence conjecture is presented as 'checked' without a single same-variety comparison. read the letter →

arxiv 2508.21268 v1 pith:XEHLMO76 submitted 2025-08-29 math.GR math.GT

Anthony Christiana , Ben Clingenpeel , Huizheng Guo , Jinseok Oh , Jozef H. Przytycki , Anna Zamojska-Dzienio This is my paper · ORCID
classification math.GRmath.GT MSC 20N0557K10
keywords quasigroupBol-Moufangtypehomologyaffineboundaryoperatorrootedbinarytreeabelianizationloop
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to start a homology theory for quasigroups of Bol-Moufang type, the class of quasigroups defined by one identity equating two different bracketings of a four-letter word. The authors derive second and third boundary operators, $\partial_2$ and $\partial_3$, by studying extensions of a quasigroup by an affine quasigroup of the same type. They prove these operators form a chain complex for every admissible choice of affine parameters $(t, s)$, so that $H_1$ and $H_2$ are defined. They then compute $H_1$ and $H_2$ for the nineteen quasigroups that Phillips and Vojtechovsky used to separate the 26 varieties, and they conjecture that $H_2$ depends only on the variety, not on which defining identity is used. If correct, this gives a new algebraic invariant attached to each Bol-Moufang variety, with $H_1(X; 1, 1)$ recovering the universal abelian quotient of the quasigroup.

What carries the argument

The central objects are the rooted binary tree polynomials $h(T)$, $H(T)$, and $Q(T)$. For a tree $T$ with ordered leaves, $h(T)$ records the path weights (words in $t$ and $s$) from the root to each internal vertex, $H(T)$ evaluates the tree as an affine word in the leaf labels $a_1, \ldots, a_n$, and $Q(T)$ records, for each internal vertex, the pair of subwords being multiplied, weighted by the path word. The boundary maps are defined by $\partial_2(x, y) = tx + sy - xy$ (obtained from equivalence of extensions) and $\partial_3(x, y, z) = Q(V^i) - Q(V^j)$ (obtained from the two trees in the Bol-Moufang identity $V^{ij}$). The proof that these form a chain complex uses the cancellation identity $\partial_2(Q(T)) = -\text{root word} + \text{sum of leaf terms}$, whose

What would settle it

Take the smallest quasigroup in the RG1L variety that satisfies both defining identities A25 and D25, compute $H_2(X; A25, 1, 1)$ and $H_2(X; D25, 1, 1)$, and compare the two abelian groups. If they are not isomorphic, Conjecture 6.1 is false. The paper's own data never performs this comparison for two identities that define the same variety, so this single computation would settle the central open claim.

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Extended reading notes

Core claim

For any quasigroup $X$ of Bol-Moufang type and any admissible substitution $(t, s)$ of automorphisms of an abelian group, the sequence $0 \to kX^3 \xrightarrow{\partial_3} kX^2 \xrightarrow{\partial_2} kX \to 0$ is a chain complex, where $\partial_2(x, y) = tx + sy - xy$ and $\partial_3(x, y, z) = Q(V^i) - Q(V^j)$ for the defining identity $V^{ij}$. Here $Q$ is a polynomial associated to a rooted binary tree that records, for each internal vertex, the pair of subwords being multiplied and the path weights in $t$ and $s$. The paper verifies $\partial_2 \partial_3 = 0$ using the identity $\partial_2(Q(T)) = -\text{root word} + \text{sum of leaf terms}$, together with the previously determined affine solutions (List 3.9). This makes $H_1$ and $H_2$ well-defined for every Bol-Moufang quasigroup. The paper further shows t

Load-bearing premise

The claim that $H_2$ depends only on the variety, not on which defining identity is chosen, is supported only by the computed examples and not proved in general; if two identities defining the same variety gave different $H_2$, the table of homology by variety would lose its meaning.

Editorial extensions

If this is right

  • Every quasigroup of Bol-Moufang type now has well-defined homology groups H₁ and H₂ for each admissible substitution (t, s), with H₁(X; 1, 1) the universal abelian quotient of X.
  • The homology H₁ for the substitutions (1, −1) and (−1, 1) is the abelianization of the parastrophes (X, /) and (X, \), respectively, giving a homology-theoretic interpretation of parastrophe duality.
  • If Conjecture 6.1 holds, H₂(X; Vⁱʲ, t, s) is an invariant of the variety membership of X rather than of the chosen defining identity, so one may speak of, for example, the 'RG1 homology' of a quasigroup.
  • For groups, the conjecture predicts that any Bol-Moufang identity defining the variety of groups yields the usual group homology of X.
  • The construction is deliberately low-dimensional (only ∂₂ and ∂₃); the paper leaves open the problem of defining higher boundary maps ∂ₙ for n ≥ 4.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If variety invariance of H₂ is confirmed, the second homology could serve as a new tool for distinguishing quasigroup varieties from each other, complementing the equational-reasoning classification of Phillips and Vojtechovsky.
  • The 'Alexander solution' t + s = 1 with c₀ = 0 that appears for the FQ0 varieties suggests a possible connection with knot-theoretic Alexander invariants and Yang-Baxter homology, where similar substitutions arise.
  • The X14 identity (four distinct variables) gives a ∂₃ whose homology is always a common quotient of the H₂ groups computed from the repeated-letter identities Aij through Fij; this quotient could be used to detect how sensitive the homology is to repetition of variables.
  • The small-category homology of the multiplication group (computed in Section 6.1) gives different invariants from the proposed Bol-Moufang homology, so the two theories capture genuinely different information about the same quasigroup.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a new (co)homology theory for quasigroups of Bol-Moufang type, based on extensions by affine quasigroups. The main construction defines a 3-term chain complex 0 → kX^3 --∂3--> kX^2 --∂2--> kX → 0, with ∂2(x,y)=tx+sy−xy and ∂3(x,y,z)=Q(T_i)−Q(T_j) for the Bol-Moufang identity V_ij. The authors verify ∂2∂3=0 in Section 4.1, prove that H1(X;1,1) is the universal abelian quotient of X, and compute H1 and H2 for the 19 distinguishing examples of Phillips and Vojtechovsky. They conjecture (Conjecture 6.1) that H2 depends only on the variety of quasigroups, not on the chosen defining identity, and they speculate about a connection to homology of small categories.

Significance. If correct, the paper provides the first systematic homology theory for Bol-Moufang quasigroups, with an explicit and computable chain complex. The construction is concrete: ∂2 is derived from equivalence of extensions, and ∂3 is defined directly from the bracketing trees, which makes the theory easy to apply. The verification of the chain complex condition is a genuine contribution, as is the interpretation of H1 as a form of abelianization, including the parastrophe cases. The extensive tables of H1 and H2 for the Phillips–Vojtechovsky examples are a useful resource. However, the central interpretive claim that H2 is a homology invariant of the variety, rather than of a quasigroup together with a chosen identity, rests on unproven Conjecture 6.1; the data presented in Section 5 do not actually test this conjecture, because no same-variety identity comparison is shown. This limits the significance of the per-variety tables until the conjecture is either proved or the presentation is restricted to the identity-specific results.

major comments (3)
  1. [List 3.9, items (12)(a) and (15)(a)] The statement that H2 is independent of the choice of defining identity within a variety is load-bearing for the organization of Section 5 and for phrases such as 'RG1 homology' in Section 6. The text says this is 'evidenced by the experimental data' (§3.4) and 'our data supports' (§5), but no displayed example compares two identities that define the same variety. Each listed quasigroup satisfies identities from distinct varieties; for instance, A1 in Example 5.1 satisfies A25, A23, B25, E25, A14, F25, C25, and A35, none of which are equivalent. To support the conjecture, one would need, e.g., a quasigroup satisfying both A25 and D25 (both RG1L) with equal H2 for the two ∂3 maps. As written, the per-variety H2 table is unsupported, and the phrase 'RG1 homology' is premature. Please either prove the conjecture, provide such same-variety comparisons, or explicitly mark all per-variety entr
  2. [List 3.9] List 3.9 contains visible errors that undermine the claim of determining the 'full set of solutions' (t,s) for every identity. In item (12)(a) (C15), the displayed h-value is h(E25), not h(C15). In item (15)(a) (A15), the solution set is listed as 's=1 and t^2+t=2', with the conclusion '(t,s)=(-2,1)'; the solution t=1 (which is always available) is omitted. Since the chain-complex verification in §4.1 relies on the completeness and correctness of these solution sets, the list must be corrected. I would also recommend that the derivation be supplemented by a computer-checkable table or script, given the number of cases.
  3. [List 3.9] The affine solution sets for all 26 varieties are summarized in List 3.9, but the derivation is carried out in detail only for the right Bol case (E25); the other 25 are said to be 'very similar' and 'left as an exercise'. Given that both the chain-complex verification and the per-variety homology calculations depend on these solution sets, this is a reproducibility gap. At minimum, the authors should include an appendix or supplementary file containing the full derivation for all cases, or a clear algorithmic description that allows the reader to verify each entry independently. The typo in C15 and the omission in A15 reinforce the need for such a check.
minor comments (4)
  1. [§3.4] The definition of H2 is written as 'H2(X; V_ij) = Im ∂3^{V_ij} / ker ∂2', which is inverted. The correct definition is ker ∂2 / Im ∂3^{V_ij}, as used in Section 5 and elsewhere. Please fix this typo.
  2. [§5] The table headers for Examples 5.7 and 5.19 read 'H2(A3; V_ij, t, s)', but the examples are for A7 and A19, respectively.
  3. [List 3.9] In the A15 entry, the expression for H(A15) appears to have a typo: the third coefficient is written as 's(s^2−1)c', while the surrounding analysis suggests it should involve t as well; please recheck.
  4. [§3.4] Notation is sometimes inconsistent, e.g., '(x, xy)t' appears where 't(x, xy)' is meant. A uniform convention for writing monomials in t and s would improve readability.

Circularity Check

1 steps flagged · score 2.0 of 10

No circular derivation: the chain complex and homology computations are self-contained; minor presentation issue: Conjecture 6.1 is used to organize per-variety tables without the same-variety comparisons that would support it.

  1. other [Section 3.4 (intro to the list of ∂3 maps), Section 5 (intro before Example 5.1), and Conjecture 6.1 in Section 6]
    "As evidenced by the experimental data, H2(X; V ij) = Im ∂V ij 3 / ker ∂2 ∼= Im ∂W kℓ 3 / ker ∂2 = H2(X; W kℓ) if V ij and W kℓ define the same variety of quasigroups; see Conjecture 6.1. Consequently the list of boundary maps that follows is organized according to the 26 equational classes (varieties) of BMq's. ... Consistent with Conjecture 6.1, our data supports H2(X, V ij, t, s) = H2(X, W kℓ, t, s) if V ij and W kℓ define the same variety of Bol-Moufang quasigroup. Thus, we show only one computation H2(X, V ij, t, s) for V ij a given representative of a variety."

    The per-variety organization of ∂3 and the Section 5 tables presupposes variety-invariance of H2: only one defining identity is used per variety, and Section 6 speaks of 'the RG1 homology.' The paper says this independence is 'evidenced by the experimental data' and 'consistent with Conjecture 6.1,' but no displayed computation compares two identities that define the same variety. Every quasigroup in Section 5 satisfies identities from distinct varieties (e.g., A1 satisfies A25, A23, B25, E25, A14, F25, C25, A35, none of which are equivalent). Thus the tables are organized by an unverified assumption presented as evidence; this is a circular-presentation/missing-support issue rather than a reduction of the homology values to their inputs.

full rationale

The central construction is not circular. ∂2(x,y)=tx+sy−xy is derived from equivalence of extensions (Proposition 3.4), and ∂3 is obtained from the cochain condition in the right Bol case and then defined generally as Q(V^i)−Q(V^j) (Definition 3.5, Section 3.4). The chain complex condition ∂2∂3=0 is verified directly in Lemma 4.1 and Corollary 4.2: the term −g(r1)+g(r2) vanishes because X satisfies the BM identity, and the remaining sum vanishes exactly for the affine (t,s) solutions tabulated in List 3.9. This is a direct computation, not an assumption of the conclusion. H1(X;1,1) as abelianization is proved by an adjunction argument (Proposition 4.3), and the H2 computations in Section 5 are explicit cokernel calculations. The only flagged weakness is Conjecture 6.1: the paper claims experimental support for H2 depending only on the variety, but no same-variety identity comparison is displayed, so the per-variety H2 tables and phrases like 'RG1 homology' rest on an unverified conjecture. This is a correctness/evidence gap, not a circular derivation. Minor typos in List 3.9 (e.g., h(E25) under C15; A15 omitting t=1 from the solution set despite (1,1) being used elsewhere) also deserve re-checking but do not affect the circularity assessment.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the choice of affine parameters (t,s), which are solved from equations rather than fitted to data, and on the correctness of the Phillips-Vojtechovsky classification. No new mathematical entities (particles, forces, dimensions) are postulated.

free parameters (2)
  • t = per variety solution set (e.g., t=±1 for RBQL; t=-2,1 for LC3L)
    Automorphism parameter of the affine quasigroup; enters ∂2(x,y)=tx+sy−xy and ∂3. Only certain values make ∂2∂3=0, solved from the affine BMq conditions in List 3.9.
  • s = per variety solution set (e.g., s=1 for RBQL; s=1 for LC3L)
    Automorphism parameter of the affine quasigroup; enters ∂2 and ∂3. Its allowed values are determined by requiring the affine quasigroup to satisfy the same BM identity.
assumptions (4)
  • domain assumption t and s commute as scalar coefficients in the chain complex
    Section 4.1 states 'we work here with the assumption that they commute for simplicity.' The paper claims commutativity follows from the affine analysis in all cases, with a no-zero-divisors assumption in four cases, but the chain complex verification adopts it directly.
  • domain assumption The coefficient ring has no zero divisors in cases A14, B23, F25, C15
    List 3.9: commutativity of t and s is only proven assuming no zero divisors in these four cases; the homology computations use the resulting solution sets.
  • domain assumption The classification of the 26 BMq varieties and the list of defining identities from Phillips-Vojtechovsky are correct
    The paper relies on [PhVo1] for the equivalence of identities and the naming of each variety; if that classification were wrong, the organization of ∂3 maps and the conjecture would be affected.
  • standard math Standard algebraic facts: Birkhoff's HSP theorem, Bruck-Murdoch-Toyoda theorem, Eilenberg extension theory
    Used throughout Sections 2 and 3 without proof, serving as background for the variety framework and the affine quasigroup structure.

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Pith. "Pith review of In Search of Homology for Quasigroups of Bol-Moufang Type." pith.science (2026). https://pith.science/paper/XEHLMO76

@misc{pith2026250821268,
  author       = {Pith},
  title        = {Pith review of: In Search of Homology for Quasigroups of Bol-Moufang Type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XEHLMO76}},
  note         = {Machine review of arXiv:2508.21268}
}
abstract

We initiate (co)homology theory for quasigroups of Bol-Moufang type based on analysis of their extensions by affine quasigroups of the same type. We use these extensions to define second and third boundary operations, $\partial_2(x,y)$ and $\partial_3(x,y,z)$, respectively. We use these definitions to compute the second homology groups for several examples from the work of Phillips and Vojtechovsky. We speculate about the relation between these homology groups and those obtained from a small category with coefficients in a functor.

Figures

Figures reproduced from arXiv: 2508.21268 by the authors.

Figure 3
Figure 3. Word (x1x2)(x3x4). 2.1.3. Enumeration and classification due to [PhVo1]. In [PhVo1] the au￾thors fully classified varieties of quasigroups of Bol-Moufang type. They showed that although there are 60 “different” identities of Bol-Moufang type, the exact number of such varieties is 26. This means that some of these identities define the same variety - one says then that identities are equivalent. Moreover, in [PhVo1] … view at source ↗
Figure 6
Figure 6. Word A2 : x((xy)z). By V ij, with V ∈ {A, . . . , F} and 1 ≤ i < j ≤ 5, we denote the identity whose left-hand side is V i and right hand-side is V j. For example, the identity A25 corresponds to A2 = A5, i.e. x((xy)z) = ((xx)y)z. The dual to the identity V ij is the identity V ′ j ′ i ′ which can be calculated due to the following rules: A ′ = F, B′ = E, C′ = C, D′ = D, 1 ′ = 5, 2 ′ = 4, 3 ′ = 3, with γ ′′ = γ for … view at source ↗
Figure 7
Figure 7. A rooted tree T. Lemma 3.8. Let w = a1 ∗ a2 ∗ ... ∗ an, with chosen bracketing given by a rooted binary tree Tw, be a word in the affine quasigroup (A, ∗) over the abelian group (A, +). If we write w as a linear combination of a1, ..., an, c0 and use the definition of ∗ in the affine case, then w = H(Tw) + h(Tw)c0 or shortly w = Hˆ (Tw). We use this lemma to find conditions for t, s and c0 for which the affine quasi… view at source ↗

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