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Binary partial groups

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Pith's one-line read This paper proves that every binary partial group—a set with a partially defined multiplication and inverses—can be promoted to a partial group in the modern, arbitrary-length sense, and that the two categories are equivalent on 2-skeletal

desk verdict Binary partial groups embed into Chermak partial groups, and the claimed equivalence with 2-skeletal partial groups is likely correct, but the current writeup leaves several load-bearing proof steps implicit. read the letter →

arxiv 2603.03167 v2 pith:UUVJMK7A submitted 2026-03-03 math.GR math.CT

Philip Hackney , Justin Lynd , Edoardo Salati This is my paper · ORCID
classification math.GRmath.CT MSC 20N0208A5520N9918F2018N5055U10
keywords binarypartialgroupsmagmaspregroupssimplicialsetssymmetric2-skeletalequivalenceofcategories
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

There are two historically distinct definitions of 'partial group.' The older, binary version gives a set with a partially defined multiplication and a two-sided inverse operation. The newer version keeps a whole simplicial structure: a chosen set of words of every length that are allowed to multiply, with consistency conditions. The paper shows that every binary partial group satisfies the newer definition in a canonical way, and that the binary partial groups form the largest class of such objects for which this is possible. It gives two embeddings from the binary to the modern category, and proves that the smaller one restricts to an equivalence of categories between binary partial groups and the 2-skeletal partial groups. This unifies the two traditions, so tools from one theory can be transferred to the other.

What carries the argument

The key object is the simplicial set BP built from a binary partial group P: its n-simplices are words (a1,...,an) such that every full parenthesization of the word is defined and yields the same element. This single construction is what converts the binary, pair-only operation into the arbitrary-length multiplication structure required by a modern partial group. The small embedding B' = sk2 ∘ B cuts BP down to its elements of dimension 0, 1, and 2; because a 2-skeletal symmetric set is determined by its vertices, edges, and triangles, this smaller object still carries the entire structure and gives the equivalence with the 2-skeletal partial groups.

What would settle it

A concrete falsifier would be a binary partial group P and a word w ∈ P^n such that the reversed-and-inverted word w† belongs to BP_n while w does not; then BP would not be closed under outer face maps and would not be a partial group. Equivalently, find w ∈ BP_n whose some consecutive subword (for example, deleting the first letter and inverting the remainder) is not in BP; Theorem 7's claim that w†w ∈ BP_2n would then fail.

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Extended reading notes

Core claim

The central claim is that the assignment P ↦ BP, where BP_n is the set of words in P^n whose every full parenthesization is defined and gives the same value, turns a binary partial group P into a genuine partial group in the modern sense. The outer face maps of BP come from the cancellation property of binary partial groups. The paper then defines a smaller embedding B' = sk2 ∘ B by taking the 2-skeleton in the symmetric-set model, and proves that B' gives an equivalence of categories between binary partial groups and 2-skeletal partial groups. Along the way it shows that the functor T, which extracts the underlying binary partial group from the 1-simplices of a modern partial group, is left

Load-bearing premise

The load-bearing premise is that the word set BP is closed under the operations needed for BP to be a simplicial set—specifically, that a word belongs to BP exactly when its reversed-and-inverted word does, and that consecutive subwords of BP-words are again BP-words; the paper proves only one direction of the first equivalence, so if the missing direction fails for some binary partial group the construction BP would not be a partial group.

Editorial extensions

If this is right

  • Every binary partial group carries a canonical modern partial-group structure, so results about modern partial groups apply to all binary partial groups.
  • The embedding is fully faithful, meaning no binary partial groups are identified or lost when moving to the modern setting.
  • The 2-skeletal partial groups are exactly the binary partial groups, up to equivalence of categories; the binary theory is the 2-dimensional shadow of the modern theory.
  • Every modern partial group has an underlying binary partial group on its 1-simplices, connected to the embedding by an adjunction with identity counit.
  • The known embedding of pregroups into modern partial groups is recovered as a special case.

Reading between the lines

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

  • Because the equivalence lands on 2-skeletal objects, the binary perspective is blind to constraints that only appear in higher-dimensional multiplication words (length 3 and beyond); one could test whether known non-associative binary partial groups hide their failure exactly at the level of 3-simplices.
  • The two embeddings are extremes—one records all coherent iterated multiplications, the other only pairs—so intermediate dimension-truncated constructions may produce a filtration of partial-group structures; this is a natural extension the paper does not explore.
  • If homotopy-theoretic invariants of partial groups are invariant under equivalence of categories, then binary partial groups could serve as low-dimensional combinatorial models for computations in the modern theory.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies 'binary partial groups': sets with an identity and a partially defined binary multiplication such that each element has a two-sided inverse in the sense of Definition 1. The main construction, in §3, associates to each binary partial group P a simplicial set BP whose n-simplices are words of length n for which every full parenthesization is defined and has the same value. The authors claim that BP is a Chermak partial group (Corollary 8), that the underlying 1-skeleton functor T is left adjoint to B (Theorem 10), that B is fully faithful (Corollary 11), and that the composite B' = sk2 ∘ B gives an equivalence between binary partial groups and 2-skeletal partial groups (Theorem 13). The paper also recovers the fully faithful embedding of Stallings pregroups into partial groups. The central claims are plausible and the proof strategy is standard, but the construction of BP as a simplicial set and several steps in Theorem 7 rely on an unproved subword-closure property.

Significance. If the identified gaps are filled, the paper gives a satisfying and nontrivial structural result: the classical 'binary' partial groups studied since Baer and Tamari are exactly the underlying object of a natural class of Chermak partial groups, and the category of binary partial groups is equivalent to the known category of 2-skeletal partial groups from [11]. This unifies legacy partial-group notions with the modern simplicial/symmetric-set framework and strengthens the earlier pregroup embedding of [16]. The paper is careful to separate its contribution from the prior characterization results [10, 11], which are used as tools rather than as the target conclusion. Machine-checked proofs or executable code are not provided, but the reasoning is of a standard, checkable kind. The main results are likely correct, but the proof of the key construction is too compressed in places that are load-bearing.

major comments (2)
  1. [§3, construction of BP] The proof that BP is a simplicial set is not supplied. Outer face maps d_0 and d_n require that every consecutive subword of a BP-word is again a BP-word. The text only says this is 'clear' or 'depends on the cancellation property,' but no proof is given. This subword-closure property is load-bearing: it is used not only to define the simplicial structure but also in the induction in Theorem 7, where products such as (a_{i+1} ... a_k) and (a_i ... a_j) are treated as having unique products without first proving that the corresponding subwords lie in BP. I recommend adding an explicit lemma: if w=(a_1,...,a_n) ∈ BP_n and 1 ≤ i ≤ j ≤ n, then the consecutive subword (a_i,...,a_j) lies in BP_{j-i+1}. A proof is short: for every full parenthesization ρ of the subword, the parenthesization obtained by multiplying on the left by a_1 (and similarly on the right by a_n) is a full parenthesization
  2. [§3, Lemma 6] Lemma 6 states that w ∈ BP_n implies w† ∈ BP_n. The proof as written has a gap: the 'preliminary consideration' proves only the forward implication 'if μ(v) is defined, then μ(v) = μ̄(v†)†', but the proof of Lemma 6 uses the converse direction when it writes μ1(v) ≍ μ̄1(w)†. The undefined/defined cases are not covered by the induction given. This is repairable: since w ∈ BP_n, μ̄1(w) is defined; applying the forward implication to the word w with parenthesization μ̄1 gives μ̄1(w) = μ1(w†)†, which both shows that μ1(w†) is defined and identifies its value. I recommend stating this explicitly and ideally proving the biconditional 'μ(v) is defined iff μ̄(v†) is defined' as a named lemma. Because Lemma 6 supplies the involution on BP and is needed for Corollary 8, this should not be left as an implicit step.
minor comments (5)
  1. [Theorem 7 statement] Typo: 'If w ∈ BP_n then, then w†w ∈ BP_{2n}' has a duplicated 'then'.
  2. [§3, after definition of BP] The sentence 'It is clear that BP possesses inner face maps ... and that these again land in BP' should reference the subword-closure lemma and the cancellation argument rather than leaving the verification to the reader, especially since the same closure is needed later.
  3. [§4, 2-skeletal partial groups] The statement 'The 2-skeleton of a partial group is again a partial group' is asserted without proof or reference. Since this is used to ensure that B' lands in 2-skeletal partial groups, a proof or a precise citation would be helpful.
  4. [§4, curiosity remark] The sentence 'for each 2 < n < ∞, the functor sk_n ∘ B is also a fully faithful embedding of binary partial groups into partial groups' is stated without justification. If it is not immediate from Theorem 11 and the 2-skeleton construction, a short argument should be added.
  5. [References] In Theorem 13, the use of [11, Theorem 9] to characterize spiny 2-skeletal sets is central; please verify that the cited theorem indeed applies verbatim to partial groups, or state the needed consequence explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the equivalence is constructed directly from binary partial groups; the only concerns are prior self-citations used as tools and a fillable gap in Lemma 6, neither of which makes the derivation circular.

full rationale

The central construction B sends a binary partial group P to the simplicial set BP of words whose full parenthesizations are all defined and agree, and the equivalence of Theorem 13 is proven by direct adjunction (Theorem 10) and the observation that η′ is the identity in degrees 0–1 and is the Segal map E2 in degree 2. No fitted parameter is renamed as a prediction, and the definition of a binary partial group is not stated in terms of Chermak partial groups. The paper does rely on two results from the authors' prior work: [10, Thm 4.6] (spiny symmetric sets vs. edgy simplicial sets) and [11, Thm 9] (2-skeletal spiny iff E2 injective). These are self-citations, but they are not circular reductions: they are published, parameter-free characterizations of the ambient symmetric-set formalism whose statements do not mention binary partial groups or the target embedding. The proof of Lemma 6 does contain a logical gap: the preliminary in §3 proves only the forward implication 'if μ(v) is defined then μ(v)=μ̄(v†)†', while the displayed chain μ1(v)≍μ̄1(w)†≍μ̄2(w)†≍μ2(v) also uses the converse to conclude that μ1(v) is defined. This is a repair needed for BP to be a simplicial set, but it is an omitted-implication proof gap, not an equivalence by construction or a self-citation. Since the paper is otherwise self-contained and the gap is fillable, the circularity score is low.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters. The constructions depend on prior theorems in simplicial/symmetric sets, two by the same authors, listed here. The two ad_hoc entries are facts used implicitly but not stated/proved as separate lemmas.

assumptions (7)
  • domain assumption Equivalence between Chermak partial groups and reduced edgy simplicial sets satisfying the inversion conditions
    Invoked in §1 to identify BP as a partial group via the † conditions rather than Chermak's original definition.
  • domain assumption [10, Theorem 4.6]: edgy simplicial sets admitting a simplicial involution † with stated conditions correspond to spiny symmetric sets
    Basis for the simplicial-set description of partial groups used throughout the paper.
  • domain assumption [11, Theorem 9]: a 2-skeletal symmetric set is spiny iff E2 is injective
    Used in Theorem 13 to identify 2-skeletal partial groups among symmetric sets.
  • standard math [19, Lemma 4.5.13]: a functor with an isomorphic counit is fully faithful
    Used in Corollary 11 to conclude B is fully faithful from the adjunction.
  • standard math Induction on word length and finiteness of parenthesizations, standard set theory
    Used throughout the constructions of BP and in the double induction of Theorem 7.
  • ad hoc to paper Consecutive subwords of BP-words are BP-words
    Used implicitly in Theorem 7 to treat expressions like a_{j+1}...a_k as valid BP-words; not stated or proved as a separate lemma.
  • ad hoc to paper For a parenthesization µ, µ(v) is defined iff ¯µ(v†) is defined, with equality µ(v)=¯µ(v†)†
    Lemma 6 uses this as a biconditional, though the preliminary paragraph only proves the forward direction.

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Pith. "Pith review of Binary partial groups." pith.science (2026). https://pith.science/paper/UUVJMK7A

@misc{pith2026260303167,
  author       = {Pith},
  title        = {Pith review of: Binary partial groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UUVJMK7A}},
  note         = {Machine review of arXiv:2603.03167}
}
read the original abstract

There are many examples of `binary' partial groups in the literature: sets equipped an identity and a partially-defined binary operation, such that each element admits an inverse. We show that many of these may be regarded as partial groups in the sense of Chermak, and single out the largest class of such objects.

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