Strong G-schemes and strict homomorphisms

Frank a Campo

Strong G-schemes and strict homomorphisms

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For finite posets, the strong G-scheme preorder is exactly the strict-homomorphism count preorder.

desk verdict A sound and genuine result in finite poset combinatorics: the equivalence between strong G-schemes and strict homomorphism inequalities is new and the proof survives scrutiny, though its significance is modest and its dependence on earlier arXiv preprints is a minor concern. read the letter →

arxiv 1908.06897 v1 pith:52LTLFX6 submitted 2019-08-19 math.CO

Frank a Campo This is my paper · ORCID

classification math.CO MSC 06A0706A06

keywords finiteposetsorderhomomorphismsstrictG-schemesHom-schemesposetpreorderEV-systemscancellationofexponents

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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 proves a characterization of the strong G-scheme preorder on finite posets: a strong G-scheme from R to S exists exactly when every finite poset P sends no more strict order homomorphisms to R than to S, and it is enough to check connected P. Strict homomorphisms are the maps that preserve strict comparabilities, so the result says that a regularity-preserving injective comparison of full homomorphism sets is controlled entirely by these stricter map counts. The characterization turns an infinite family of existence questions into cardinality comparisons, and it yields the corollary that two finite posets with equal strict-homomorphism counts for every finite P are isomorphic. The paper also supplies a finite sufficient condition for the relation and a construction producing new examples of the form P + Q ⊑_G P|A + T, where A is a convex subposet of P.

What carries the argument

The load-bearing object is the quotient partition G(ξ) of a homomorphism ξ: P → Q: its blocks are the connected components Gξ(x) of the preimage $ξ^{{-1}}$(ξ(x)) containing x. Every homomorphism factors as ξ = ιξ ∘ πξ, with πξ collapsing each block to a point and ιξ a strict homomorphism from the quotient poset G(ξ) to Q. The key counting lemma states that for a fixed quotient G(ξ), the homomorphisms from P to T sharing that quotient are in bijection with the strict homomorphisms from G(ξ) to T, so #Γ_{P,T}(ξ) = #S(G(ξ),T). This identity converts the strict-count inequality for the arbitrary test poset G(ξ) into the fiberwise injectivity needed to build a strong G-scheme.

What would settle it

Enumerate all pairs of finite connected posets up to, say, eight points and compute #S(P,R) and #S(P,S) for all connected P up to the same size; a pair where the inequalities all hold but an explicit search for a strong G-scheme finds none would refute the theorem, while a reversed inequality would show R ⊑_G S fails.

Watch

Extended reading notes

Core claim

The paper's main theorem states that for finite posets R and S the following are equivalent: a strong G-scheme from R to S exists; #S(P,R) ≤ #S(P,S) for every finite poset P; and #S(Q,R) ≤ #S(Q,S) for every connected finite poset Q. A strong G-scheme is a family of injective maps from the homomorphism sets H(P,R) into H(P,S), one family member for each isomorphism type P, that preserves the connected-component structure of each homomorphism's preimage fibers. The paper thereby reduces a regular injective comparison of all homomorphism sets to a plain numerical comparison of strict-homomorphism counts. An immediate consequence is that #S(P,R) = #S(P,S) for all finite P forces R ≅ S; the paper further derives a finite-check sufficient condition for the preorder and a construction method for posets T with P + Q ⊑_G P|A + T for convex A.

Load-bearing premise

The proof imports from earlier work the fact that a homomorphism is strict exactly when every point is isolated within the connected component of its own preimage, together with a lemma on when those components grow; the equivalence between strong G-schemes and strict-homomorphism count inequalities breaks if these imported facts fail.

Editorial extensions

If this is right

The preorder on finite posets defined by strong G-schemes is the same as pointwise comparison of the sequences (#S(P,R))_P, so all structural facts about the G-scheme preorder can be read off strict-homomorphism counts.
To decide R ⊑_G S, only connected test posets need to be checked; disconnected P factor as products over components.
If all finite posets P give #S(P,R) = #S(P,S), then R and S are isomorphic; this refines the classical homomorphism-count cancellation result to strict maps.
A finite certificate suffices in many cases: Theorem 2 reduces the infinite check to a finite set of connected posets, embedding counts, and distributors.
Theorem 3 constructs new pairs P + Q ⊑_G P|A + T whenever A is convex in P, giving a systematic source of nontrivial strong G-schemes.

Reading between the lines

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

The quotient factorization suggests defining the strict-homomorphism profile of a poset as the vector of counts #S(P,·) over connected P; the paper shows this profile completely determines the G-scheme preorder, so it may serve as a complete invariant analogous to homomorphism-count profiles elsewhere.
Because Theorem 1 makes the G-scheme relation a cardinality comparison, finite-precision obstructions can be sought by computing strict counts only, which is plausible for computer enumeration; the paper does not address complexity or bounds.
A testable extension is to ask whether the same strict-count characterization holds for other relational structures whose fibers have a connectivity notion, such as graphs with zigzag-connected fibers under graph homomorphisms; the paper does not claim this.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main result is real. Theorem 1—that R ⊑_G S iff #S(P,R) ≤ #S(P,S) for every finite poset P, with the connected version equivalent—is a clean characterization and I don't see it in the cited literature. The corollary extending Lovász's counting theorem to strict homomorphisms is a natural and correct byproduct. The proof is also in better shape than I expected: the quotient construction on G(ξ), Lemma 5's bijection between the fibers and strict maps out of the quotient, and the use of Lemma 2 all check out. I walked through the key implication (2)⇒(1) step by step, including the factorization argument and the injectivity claim, and it holds. The self-citations to [5] for the strictness criterion and Lemma 2 are not a serious risk; as the stress-test note says, those facts are elementary once you have connectedness of the G-fibers.

What the paper does well beyond the theorem: it gives a finite-checkable sufficient condition (Theorem 2) and a construction method (Theorem 3), and it tests both on worked examples. The two examples in Section 5.1 are genuinely illustrative, not decorative. This is a toolbox-oriented contribution, and the tools are nontrivial.

Soft spots are real but mostly minor. The paper leans heavily on the author's prior work [5, 6], which are arXiv preprints; a reader without those in hand has to take several foundational definitions and results on faith. That is acceptable in a niche subfield, but an editor should ask for the imported lemmas to be stated clearly enough to be checked without constant cross-referencing. The presentation is dense, with a few typos and some notation that takes getting used to, especially the EV-system calculations. The significance is confined to finite poset theory and the author's research program; the impact is not broad. But that is not a flaw—it's just scope.

The citation pattern is honest. Lovász, Birkhoff, Hashimoto, McKenzie, Duffus, and Wille are cited in the right places. The author's self-citations are extensive but appropriate given the subject; the key novelty is the strict-homomorphism characterization, which is not in the earlier papers.

Who is this for? Specialists in ordered sets and homomorphism-counting questions. It deserves a serious referee: the central theorem is correct, the proof is original, and the paper moves a longstanding question—what conditions on homomorphism sets force order-isomorphism—in a new direction. A competent referee can verify the main line without too much pain, and the added tools are checkable. Recommend sending it to peer review, with the caveat that the author should be asked to make the paper more self-contained with respect to [5, 6] before publication. I would cite the main theorem in future work on poset homomorphism counts.

Referee Report

0 major / 5 minor

Summary. The paper studies the preorder R ⊑_G S defined by the existence of a strong G-scheme from R to S. The main result, Theorem 1, characterizes R ⊑_G S by the strict-homomorphism-counting inequalities #S(P,R) ≤ #S(P,S) for all finite posets P, and equivalently for all connected finite posets P. The proof factorizes a homomorphism ξ through G(ξ), the partition of the carrier into connected components of the fibers of ξ. Corollary 1 shows G(ξ) is a poset with a strict natural map to the target; Lemma 5 identifies the fiber Γ_{P,T}(ξ) with the set S(G(ξ),T) of strict homomorphisms, which yields the hard direction (2)⇒(1) of Theorem 1. Corollary 3 deduces that equality of all strict-homomorphism counts forces isomorphism. Theorem 2 provides a sufficient condition for R ⊑_G S based on finitely many connected posets and distributors, and Section 5 applies it to two examples and develops Theorem 3, a construction for posets P+Q and T with P+Q ⊑_G P|A + T, where A is convex in P, plus a strong I-scheme strengthening when A is an antichain.

Significance. If Theorem 1 is correct, it is a clean structural characterization: the seemingly more complicated G-scheme preorder is equivalent to a monotonicity condition on strict homomorphism counts, and equality of those counts is a new proof that a finite poset is determined up to isomorphism by the cardinals #S(P,·). The reduction to connected posets and the finite criterion in Theorem 2 are useful tools. I checked the central proof carefully: the quotient construction in Corollary 1 is sound, the bijection in Lemma 5 is valid, and the use of Lemma 4 transfers the inequalities exactly as claimed. The paper is not fully self-contained, since two load-bearing facts are quoted from the author's earlier preprint [5], namely the strictness criterion Gξ(x)={x} and Lemma 2; these are elementary and are used consistently, so I do not regard the dependence as a correctness risk, only as a presentation issue.

minor comments (5)

[Section 2.2 / Definition 2 and Theorem 1] The criterion "ξ is strict iff Gξ(x)={x}" is quoted from [5, Corollary 3] and is used in the first step of the proof of Theorem 1; because this fact is load-bearing for the main equivalence, please include a short proof or at least a fully explicit statement so that the dependence on an external preprint is transparent.
[Lemma 4] Lemma 4 is stated for all P∈P, but the G-scheme in Definition 3 is defined only on the representation system P_r; the proof should explicitly invoke that every finite poset is isomorphic to a representative in P_r and that the sets Γ_{P,R}(ξ) and Γ_{P,S}(ξ) are invariant under such isomorphisms.
[Theorem 3 proof] In the embedding-counting inequality at the end of the proof, the step #Emb(E,A') + #(F1∪F2) ≥ #F1 + #F2 is compressed; it follows by inclusion-exclusion from the injection F1∩F2 → Emb(E,A') constructed in the preceding sentence, and that derivation should be written out explicitly.
[Section 5.1] The sentence "the two outer ones in E(C3;0011)" is not self-explanatory; please annotate Figure 6 or describe the two points explicitly so the claimed distinguishing property can be checked without reading the figure labels in a particular way.
[Throughout] There are several typographical slips that should be corrected, including "dubble-N" for double-N in Section 2.1, "poests" in the introduction, "fullﬁlls" in Section 5.1, and the German-size remnants "Größe 60%" and "Größe 45%" in the figure captions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is proven by a self-contained factorization argument; self-citations are independent prior lemmas, not inputs in disguise.

full rationale

The central result, Theorem 1, is not assumed from prior work but is proven here. The key implication (2)⇒(1) proceeds by factorizing a homomorphism ξ through the quotient poset G(ξ) of its connected Gξ-fibers. Lemma 5 establishes a genuine bijection between Γ_{P,T}(ξ), the homomorphisms sharing the same fiber partition as ξ, and strict homomorphisms S(G(ξ),T), and this bijection is verified rather than postulated. The strict-homomorphism inequality (2) is then applied to the finite poset G(ξ), which is legitimate because (2) ranges over all finite posets. The only imported facts are [5, Corollary 3] (strictness is equivalent to singleton Gξ-fibers) and [5, Lemma 1] (a technical comparison of fiber partitions); both are elementary, parameter-free statements whose assumptions do not include Theorem 1, and neither is equivalent to the target inequality. Corollary 3 uses the self-cited antisymmetry of ⊑G from [5], but this is a prior result about G-schemes, not the strict-homomorphism cardinality equality being derived, so its use is a normal reliance on earlier work rather than a circular reduction. No equation in the paper is shown by construction to be identical to an input, and no fitted parameter is renamed as a prediction. The proof is self-contained in its main derivation, and the self-citations identified are not load-bearing in a way that would make the conclusion equivalent to its premises.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

Pure mathematics with no fitted constants. The central theorem is proven from the paper's definitions plus several quoted results from the author's own earlier papers ([5],[6]) and from Lovasz [18]; these citations supply the strictness characterization, the partial-order property of the G-scheme relation, and the factorization formula.

assumptions (5)

domain assumption A homomorphism xi in H(P,Q) is strict iff G_xi(x) = {x} for all x in P.
Quoted from [5, Corollary 3], used in the proof of Theorem 1 to show a G-scheme maps strict homomorphisms to strict homomorphisms and in Lemma 5 to prove the converse.
domain assumption If G_xi(x) is a subset of G_zeta(x), then G_xi(x) is a proper subset iff there exist a,b in G_zeta(x) with a < b and xi(a) < xi(b).
Quoted from [5, Lemma 1], used in Lemma 5 to prove that zeta = sigma composed with pi_xi has the same G partition as xi.
standard math Lovasz factorization: for connected P, #I_Q(P,T) = #So_r(P,Q) * #Emb(Q,T), where I_Q(P,T) is the set of strict homomorphisms with image isomorphic to Q.
Cited from Lovasz [18] and used to prove Theorem 2 and Corollary 4.
domain assumption The G-scheme relation is a partial order on the representation system Pr.
Cited from [5, Theorems 2 and 3], used in Corollary 3 to turn mutual strong G-schemes into isomorphism.
domain assumption E(P+Q) = E(P) + E(Q) for EV-systems.
Quoted from [6] and used in the final I-scheme construction in Section 5.2.

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Pith. "Pith review of Strong G-schemes and strict homomorphisms." pith.science (2026). https://pith.science/paper/52LTLFX6

@misc{pith2026190806897,
  author       = {Pith},
  title        = {Pith review of: Strong G-schemes and strict homomorphisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/52LTLFX6}},
  note         = {Machine review of arXiv:1908.06897}
}

abstract

Let $\mathfrak{P}_r$ be a representation system of the non-isomorphic finite posets, and let ${\cal H}(P,Q)$ be the set of order homomorphisms from $P$ to $Q$. For finite posets $R$ and $S$, we write $R \sqsubseteq_G S$ iff, for every $P \in \mathfrak{P}_r$, a one-to-one mapping $\rho_P : {\cal H}(P,R) \rightarrow {\cal H}(P,S)$ exists which fulfills a certain regularity condition. It is shown that $R \sqsubseteq_G S$ is equivalent to $\# {\cal S}(P,R) \leq \# {\cal S}(P,S)$ for every finite posets $P$, where ${\cal S}(P,Q)$ is the set of strict order homomorphisms from $P$ to $Q$. In consequence, $\# {\cal S}(P,R) = \# {\cal S}(P,S)$ holds for every finite posets $P$ iff $R$ and $S$ are isomorphic. A sufficient condition is derived for $R \sqsubseteq_G S$ which needs the inspection of a finite number of posets only. Additionally, a method is developed which facilitates for posets $P + Q$ (direct sum) the construction of posets $T$ with $P + Q \sqsubseteq_G A + T$, where $A$ is a convex subposet of $P$.

Figures

Figures reproduced from arXiv: 1908.06897 by the authors.

**Figure 1.** The posets A2, C3,Λ3, V3, N, W and N(2) . for all x, y ∈ X, and it is called an embedding iff ξ(x) ≤Q ξ(y) ⇒ x ≤P y for all x, y ∈ X. Finally, an embedding is called an isomorphism iff it is onto. P ' Q indicates isomorphism. For posets P and Q, we use the following symbols for homomorphism sets: H(P, Q) ≡ {ξ : X → Y | ξ is a homomorphism from P to Q} , S(P, Q) ≡ {ξ ∈ H(P, Q) | ξ is strict } , S o (P, Q) ≡ {ξ ∈ H(P,… view at source ↗

**Figure 2.** Three pairs of posets R and S with R vG S. For a poset P with connectivity components Q1, . . . , QL, L ∈ N, we have for every R ∈ P H(P, R) ' H(Q1, R) × · · · × H(QL, R), and similar for S(P, R). A subset A ⊆ P is called connected (in P) iff γA(x) = A for an (arbitrary) x ∈ A. For posets P and Q, P connected, the image ξ(P) of P under a homomorphism ξ ∈ H(P, Q) is connected in Q; in particular, ξ(P) is a subset of … view at source ↗

**Figure 3.** The posets C2 + C2, A1 + Λ3, N, and C3 + A1 and their EV-systems. The points of the posets are labeled by binary words. In the EV-systems, the sets E(P; x) with x ∈ P are encircled and labeled with the respective x. Definition 5. Let P, Q ∈ P, and let ξ ∈ H(P, Q) be a strict homomorphism. We define for every x ∈ P αP,ξ(x) = ξ(x), ξ(↓ ◦ x), ξ(↑◦ x) . If P is fixed, we write αξ(x) instead of αP,ξ(x). According to … view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: For the posets R = N, S = A1 + C3 in the upper part and for the posets R = W, S = A1+N(2) in the lower part, the following matrices are shown (from left to right, zeros are omitted): (#S o r (Q, Q0 )) with Q, Q0 ∈ E c (R)∪E c (S), (# Emb(Q, T)) with Q ∈ E c (R) ∪ E c (…

**Figure 5.** Figure 5: Two pairs of posets R and S with R vG S. 45 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: Top: τ1 ∈ So (V3, C3) and ατ1 ∈ S(V3, E(C3)). Middle: τ2 ∈ S o (Λ3, C3) and ατ2 ∈ S(Λ3, E(C3)). Bottom: τ3 ∈ So (N, C3) and ατ3 ∈ S(N, E(C3)). 5.1 Proving R vG S by means of Theorem 2 Let R = N and S = A1 + C3, as shown in [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: Top: τ1 ∈ So (N, N(2)) and ατ1 ∈ S(N, E(N(2))). Middle: τ2 ∈ S o (N, N(2)) and ατ2 ∈ S(N, E(N(2))). Bottom: τ3 ∈ So (W, N(2)) and ατ3 ∈ S(W, E(N(2))). 5.2 A construction method based on Corollary 4 70 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

**Figure 8.** Figure 8: Three pairs of posets Ri , Si with Ri vG Si . In [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: Illustration of a method for the construction of posets [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [5]

Generalized One-to-One Mappings between Homomorphism Sets of Digraphs

F. a Campo: About generalized one-to-one mappings between sets of order homomorphisms. arXiv:1906.11758v2 [math.CO]

work page Pith review arXiv 1906

[2] [1]

Bergman, R

C. Bergman, R. McKenzie, and Z. Nagy: How to cancel a linearly ordered exponent. Coll. Math. Soc. J. Bolyai 29 (1977), 87–93

work page 1977

[3] [2]

Birkhoﬀ: An Extended arithmetic

G. Birkhoﬀ: An Extended arithmetic. Duke Math. J. 3 (1937), 311–316

work page 1937

[4] [3]

Birkhoﬀ: Generalized arithmetic

G. Birkhoﬀ: Generalized arithmetic. Duke Math. J. 9 (1942), 283–302

work page 1942

[5] [4]

a Campo: Relations between powers of Dedekind numbers and expo- nential sums related to them

F. a Campo: Relations between powers of Dedekind numbers and expo- nential sums related to them. J. Int. Seq. 21 (2018), Article 18.4.4

work page 2018

[6] [6]

Calculation Rules and Cancellation Rules for Strong Hom-Schemes

F. a Campo: Calculation rules and cancellation rules for strong Hom- schemes. arXiv:1908.05681 [math.CO]

work page Pith review arXiv 1908

[7] [7]

M. M. Day: Arithmetic of ordered sets. Trans. Amer. Math. Soc.58 (1945), 1–43

work page 1945

[8] [8]

Duﬀus: Powers of ordered sets

D. Duﬀus: Powers of ordered sets. Order 1 (1984), 83–92. 22

work page 1984