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Planar semilattices and nearlattices with eighty-three subnearlattices

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Pith's one-line read A high subnearlattice count forces a planar diagram

desk verdict A sharp nearlattice analogue of Czedli's 83-sublattice theorem, built on a clever new qn-lattice technique; the proof is credible but rests on unverified computer output and contains one false lemma that needs checking. read the letter →

arxiv 1908.08155 v1 pith:EQHVVOXL submitted 2019-08-22 math.RA

Gábor Czédli This is my paper · ORCID
classification math.RA MSC 06A1206B7520M10
keywords planarnearlatticesemilatticesubnearlatticecountrelativenumberofsubuniversesqn-latticecomputer-assistedproofsharpthresholdcommutativeidempotentsemigroup
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 proves a sharp counting threshold for planarity in finite nearlattices: any $n$-element nearlattice with at least $83\cdot 2^{n-8}$ subnearlattices must have a planar Hasse diagram. For every $n\ge 9$ the bound is sharp, because a non-planar $n$-element nearlattice exists with exactly $83\cdot 2^{n-8}-1$ subnearlattices. Finite nearlattices include finite semilattices, commutative idempotent semigroups, join-enriched meet semilattices, and chopped lattices, so the result transfers to all these forms. The proof works by assuming a smallest counterexample, encoding small configurations as qn-lattices, and using computer-generated subalgebra counts to exclude every possible configuration.

What carries the argument

Two devices carry the argument. The first is the relative number of subuniverses, $\sigma(L)=|\operatorname{Sub}(L)|\cdot 2^{8-n}$, which normalizes the count so that the threshold becomes $\sigma>83$ independent of $n$. The second is the notion of a qn-lattice: a finite poset with partial join and meet operations satisfying five closure axioms, understood as a weak partial subalgebra of a nearlattice. A monotonicity lemma shows that every sub-qn-lattice has $\sigma$-value at least that of the ambient nearlattice, so a minimal counterexample cannot contain any qn-lattice of $\sigma\le 83$. The proof then parses the possible shapes of a minimal counterexample into finitely many cases, computes each case's $\sigma$-value with the program 'sublatts', and uses geometric lemmas about pointed contours and cells to glue planar pieces whenever a configuration would otherwise be unavoidable.

What would settle it

Recompute every $\sigma$-value printed in the output files of Sections 5 to 7 by running 'sublatts' on the corresponding input files; if any value reported as at most 83 is actually larger, or if any printed output does not match the cited case, the exclusion argument breaks.

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Extended reading notes

Core claim

Finite upper nearlattices are join-semilattices whose meet is defined exactly when the two elements have a common lower bound; equivalently they are finite semilattices or commutative idempotent semigroups. The main theorem states that if such an $n$-element structure has at least $83\cdot 2^{n-8}$ subnearlattices, namely nonempty subsets closed under the total join and the partial meet, then its Hasse diagram is planar. Equivalently, writing $\sigma(L)=|\operatorname{Sub}(L)|\cdot 2^{8-n}$, every finite nearlattice with $\sigma(L)>83$ is planar. For $n\ge 9$ the statement is sharp: there exists an $n$-element non-planar nearlattice with $\sigma(L)=83$, that is, with exactly $83\cdot 2^{n-8}-1$ subnearlattices. Since finite lattices are nearlattices with a smallest element, the theorem generalizes the earlier planar-lattice result, and the proof reduces the lattice case to that prior result.

Load-bearing premise

The argument stands on the correctness of the computer program 'sublatts' and on the faithful transcription of every case tree into its input files; a single wrong reported count or a missed subcase in Sections 5 to 7 could let a minimal counterexample survive.

Editorial extensions

If this is right

  • Every finite semilattice, viewed as a nearlattice, has a planar Hasse diagram when its subnearlattice count reaches the $83\cdot 2^{n-8}$ threshold.
  • For each $n\ge 9$, the constant $83\cdot 2^{n-8}$ cannot be lowered while preserving the implication, since a non-planar example sits exactly one subnearlattice below the threshold.
  • The theorem extends the planar-lattice threshold to all finite commutative idempotent semigroups and chopped lattices, not only lattices with a least element.
  • At the smallest sizes the threshold is even stronger in some cases: every nearlattice with at most seven elements is planar, and every eight-element nearlattice with at least 74 subnearlattices is planar.

Reading between the lines

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

  • The constant 83 is a computed boundary rather than a value tied to an obvious structural family; a structural description of nearlattices sitting just below the threshold might explain why 83 is the extremal number.
  • Because the program's input files are not bundled, an independent reimplementation of 'sublatts' or a machine-checked translation of the case trees would turn the computer-assisted portion of the proof into a fully verifiable computation.
  • The qn-lattice closure construction is general enough that the same normalization-plus-exclusion scheme may yield sharp thresholds for other classes of partial algebras, such as algebras with one partial operation of higher arity.
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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 / 4 minor

Summary. The paper proves a sharp threshold for planarity of finite nearlattices: an n-element nearlattice with at least 83·2^(n−8) subnearlattices must be planar, and for every n ≥ 9 there exists a non-planar n-element nearlattice with exactly 83·2^(n−8)−1 subnearlattices. The proof works with a minimal counterexample and introduces qn-lattices and jm-constraints as technical tools. It then excludes a large finite family of configurations by combining structural lemmas about entries, anchors, and bridges with σ-values computed by the program 'sublatts'. The sharpness part is inherited from the author's earlier theorem for lattices.

Significance. If the result is correct, it is a natural nearlattice analogue of Czédli's theorem on lattices with 83·2^(|L|−8) sublattices and strengthens the author's subsequent semilattice result with threshold 127·2^(n−8). The qn-lattice/jm-constraint framework is a potentially reusable method for reducing subalgebra-counting questions to finite combinatorial checks. The paper is well organized, the minimal-counterexample argument is clearly structured, and the geometric lemmas in Sections 3 and 8 are presented in detail. The main weakness is that the numerical core of the proof is not independently verifiable from the submission: the program is not bundled, the input files are not supplied, and one lemma used to justify the σ-values is false as stated.

major comments (2)
  1. [Section 4, Lemma 4.7] The K-independence claim of Lemma 4.7 is false. Let W = {a∨b = c} over KW = {a,b,c}, and let K = KW ∪ {d} with a ≤ d ≤ c in the ambient nearlattice. Axiom (A5) forces d∨b = c, so the subset {b,d} is not a subuniverse of K; consequently |Sub(K)| is not 2^(|K\KW|)·|Sub(KW)| and σ(K) ≠ σ(KW). Since Lemma 4.8 is stated as a consequence of Lemmas 4.2 and 4.7, the proof of the main exclusion mechanism needs repair. The repair is local: define σ(W) directly as σ(KW) and derive Lemma 4.8 from Lemma 4.2 alone; the false independence statement and the remark about input files containing extraneous elements should then be removed or amended.
  2. [Sections 4–7 and Appendix 2] The contradiction in the proof of Theorem 2.2 rests on roughly one hundred σ-values computed by the program 'sublatts' (for example, Lemma 5.5 uses ten values and Lemmas 7.13–7.17 use values such as 77, 71.75, 69.5, 75, 74, 78.75, 79, 70.5, 82, and 81.5). The program is not included in the arXiv submission, the input files are not provided (only one sample appears in Section 9), and no machine-checked certificate is supplied. These values are load-bearing: via Lemma 4.8 each value σ ≤ 83 excludes a case, and the final contradiction requires that every listed case really is excluded. Without the program, all input files, and a reproducible way to regenerate the printed outputs, the central claim is credible but unverified by the submitted material. Please supply the program, the complete set of input files, and a script or certificate that reproduces the σ-values in Appendix 2.
minor comments (4)
  1. [Section 3, proof of Lemma 3.4] The sentence 'Since p′(0) > 0 by (3.8)' should refer to p(0) > 0; equation (3.8) establishes the positivity of p at 0, not of p′ at 0.
  2. [Section 9, Algorithm 9.1] The input-file syntax is documented only by a running example; a short formal description of the commands and constraints syntax would substantially help independent re-runs of the program.
  3. [Abstract and Theorem 1.2] The abstract says 'For n > 8, this result is sharp' while Theorem 1.2 says 'For n ≥ 9'; these are equivalent for integer n, but the wording should be unified.
  4. [Remark 1.3] The proof of the meet-reducible case in Remark 1.3 is omitted with 'analogous details are omitted'; since this remark is not used in the main theorem, this is acceptable, but a reference or a brief argument would be preferable.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation uses minimal-counterexample arguments and independent prior theorems, and the fixed threshold 83 is not fitted to the computed σ-values.

full rationale

The Main Theorem is not obtained by fitting a parameter. The quantity σ is defined in Definition 2.1 from the actual count |Sub(L;≤,∨,∧)|, and the constant 83 appears in the statement itself; it is not adjusted after inspection of the computed values. The exclusion lemmas (Lemmas 5.4–7.17, with outputs in Appendix 2) report σ-values computed by the program 'sublatts' from the stated jm-constraint inputs, and each lemma compares the result with the fixed threshold 83. These are computations from the definitions, not predictions of planarity whose truth is assumed. The proof proceeds by the standard minimal-counterexample convention (Convention 5.1): assuming Theorem 2.2 fails, it derives structural restrictions on a smallest counterexample and obtains contradictions. Lemma 5.2 invokes Theorem 2.2 for proper subnearlattices, which is legitimate because minimality supplies the inductive hypothesis. The only self-citations are the lattice analog Theorem 1.4 from Czédli [13], the semilattice analog Theorem 1.5 from Czédli [14], and Lemma 4.2, which is quoted as a particular case of Lemma 2.3 of [13]. These are prior results whose stated assumptions (finite lattices or finite semilattices with σ-thresholds) do not include the nearlattice theorem being proved; they are used as external supporting theorems, in particular to cover the lattice case and to supply the sharpness example. No uniqueness theorem is imported to forbid alternative approaches, and no ansatz is smuggled in via citation: the qn-lattice and jm-constraint formalism is introduced and proved in Section 4, and the program is described in Section 9. The absence of the executable program or a machine-checked certificate from the arXiv submission is a reproducibility and verification limitation, not a circularity; a wrong σ-value would break the proof but would not make the derivation equivalent to its inputs. Likewise, the skeptic's claim that Lemma 4.7 is false is a correctness concern about a stated independence property, not a circular-definition concern. Therefore no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper introduces no free parameters: the constant 83 is part of the theorem's content, not fitted to data. The main nonstandard assumption is the correctness of the computer-assisted enumeration used throughout the proof. The qn-lattice device is an invented mathematical structure, but it is carefully defined and used rather than pulled in ad hoc to force the result.

assumptions (5)
  • domain assumption Finiteness convention: every structure occurring in the paper is assumed finite even if not repeated.
    Stated in (1.1); ensures principal filters of nearlattices are lattices and that all chains and enumerations are finite. Used throughout the proof.
  • domain assumption For a finite join-semilattice, the structures (L;∨), (L;≤,∨,∧), (L;≤), and the partial algebra (L;∧) mutually determine each other.
    Equation (1.2); used to switch freely between join-semilattice and nearlattice viewpoints, including in Lemma 5.3 and Section 8.
  • standard math A poset has a planar curved diagram if and only if it has a straight planar diagram (Kelly's theorem).
    Equation (3.1), cited to Kelly [45]; used in Corollaries 3.5 and 3.6 to conclude planarity from curved diagrams.
  • standard math Kelly-Rival lemmas on planar lattices, including the existence of doubly irreducible elements and properties of boundaries and cells.
    Used in Remark 1.3 and in Section 8 (e.g., claims (8.7), (8.10), and the use of Proposition 1.7 of [46]); these are standard results in the theory of planar lattices.
  • ad hoc to paper Correctness of the program 'sublatts' and of the manual encoding of all jm-constraint cases in the input files.
    The contradiction proof depends on every reported σ-value in Sections 5 to 7 being correct and every case parsing tree being complete. The program is described but no certificate or machine-checked output is provided; the outputs in Appendix 2 are the only evidence.
invented entities (1)
  • qn-lattices (partial jm-algebras with ordering satisfying axioms (A1) to (A5))
    purpose: Technical substructures used to transfer upper bounds on σ via Lemma 4.2 and to enable computer enumeration of subuniverses for partial operations.
    Introduced in Definition 4.1 as a proof device. Their properties are proven internally, but they make no testable predictions outside the proof and have no independent empirical handle.

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Pith. "Pith review of Planar semilattices and nearlattices with eighty-three subnearlattices." pith.science (2026). https://pith.science/paper/EQHVVOXL

@misc{pith2026190808155,
  author       = {Pith},
  title        = {Pith review of: Planar semilattices and nearlattices with eighty-three subnearlattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EQHVVOXL}},
  note         = {Machine review of arXiv:1908.08155}
}
abstract

Finite (upper) nearlattices are essentially the same mathematical entities as finite semilattices, finite commutative idempotent semigroups, finite join-enriched meet semilattices, and chopped lattices. We prove that if an $n$-element nearlattice has at least $83\cdot 2^{n-8}$ subnearlattices, then it has a planar Hasse diagram. For $n>8$, this result is sharp.

Figures

Figures reproduced from arXiv: 1908.08155 by the authors.

Figure 1
Figure 1. A pointed contour and two rotated pointed contours Their differentiability ensures that curved edges keep going strictly upwards. Since the curved edges are the graphs of differentiable functions (but the role of the x-axis is interchanged with that of the y-axis), they have directional vectors at each of their points; note that a directional vector is of length 1 by definition. In case of a curved edge, the directi… view at source ↗
Figure 2
Figure 2. Illustration for Corollary 3.6 Corollary 3.6. Let (L; ≤, ∨, ∧) and (K; ≤, ∨, ∧) be planar nearlattices and fix a planar diagram for each of them. By taking isomorphic copies if necessary, we can assume that L∩K = ∅. Let u, v ∈ L be distinct elements on the same (left or right) boundary chain of the same cell of (L; ≤, ∨, ∧), with respect to its fixed diagram, such that u < v and u is not the smallest element of the … view at source ↗
Figure 3
Figure 3. The qn-lattices (S`; ≤, ∨, ∧) for ` ∈ {1, . . . , 4} Example 4.4. To exemplify Definition 4.3 (iv), we define four qn-lattices as follows; note that they will be needed later. Let W := {a ∨ b = m, a ∨ c = m, b ∨ c = m, d ∧ m = a, e ∧ m = b, f ∧ m = c}, W1 := {d ∨ m = i, e ∨ m = i, f ∨ m = i}, W2 := {d ∨ m = i, e ∨ m = i, f ∨ m = j}, W3 := {d ∨ m = j, e ∨ m = i, f ∨ m = j}, W4 := {d ∨ m = i, e ∨ m = j, f ∨ m = k}; se… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Three join-semilattices and two qn-lattices We are going to prove several properties of (Lce; ≤, ∨, ∧) until it appears that (Lce; ≤, ∨, ∧) cannot exist; this will imply Theorem 2.2. We begin with the following easy lemma. As usual, a subnearlattice is proper if it is …
Figure 5
Figure 5. Figure 5: The qn-lattices (T`; ≤, ∨, ∧), ` ∈ {5, 6, . . . , 10} Consider the qn-lattice (T7; ≤, ∨, ∧) determined by W∗ from (7.3) and [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: The qn-lattices (Q`; ≤, ∨, ∧), ` ∈ {1, . . . , 5} Next, let Q1 be the qn-lattice determined by [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]
Figure 7
Figure 7. Figure 7: An example for the tree of subcases Algorithm 9.1. (i) Start with “ (C) 0 be ”. Here be is the end-of-file symbol, (C) refers to the main case. The encircled 0 after the main case means that none of its subcases has been processed yet. The underlining means the positio…

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