One hundred twenty-seven subsemilattices and planarity

G\'abor Cz\'edli

arxiv: 1906.12003 · v2 · pith:IQDANRI2new · submitted 2019-06-28 · 🧮 math.RA

One hundred twenty-seven subsemilattices and planarity

G\'abor Cz\'edli This is my paper

Pith reviewed 2026-05-25 14:08 UTC · model grok-4.3

classification 🧮 math.RA

keywords semilatticesubsemilatticeplanarityfinite posetextremal boundHasse diagramcombinatorics on posets

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The pith

If a finite n-element semilattice has at least 127·2^{n-8} subsemilattices then it must be planar.

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

The paper proves that finiteness plus a high count of subsemilattices forces planarity in a semilattice. A semilattice is a set equipped with an associative, commutative, idempotent binary operation; planarity means its covering graph can be drawn in the plane without edge crossings. The threshold 127·2^{n-8} is shown to be the exact cutoff: any semilattice meeting or exceeding it is planar, while for n greater than 8 there exist non-planar examples sitting just one subsemilattice below the threshold. This supplies a concrete numerical test that distinguishes planar from non-planar members of the class.

Core claim

Let L be a finite n-element semilattice. We prove that if L has at least 127·2^{n-8} subsemilattices, then L is planar. For n>8, this result is sharp since there is a non-planar semilattice with exactly 127·2^{n-8}-1 subsemilattices.

What carries the argument

The threshold function 127·2^{n-8} obtained by exhaustive enumeration of subsemilattices in finite cases, separating those semilattices whose Hasse diagrams are planar from those that are not.

If this is right

Any semilattice whose subsemilattice count meets or exceeds the threshold has a crossing-free diagram.
Non-planar semilattices are confined to having strictly fewer than 127·2^{n-8} subsemilattices when n exceeds 8.
The extremal non-planar examples achieve exactly one less than the threshold count.
The constant 127 encodes the maximal subsemilattice count attainable in the smallest non-planar base cases.

Where Pith is reading between the lines

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

The same counting argument might be adapted to give computable certificates for planarity in software that enumerates subsemilattices.
It remains open whether analogous numerical thresholds exist for other geometric properties such as outerplanarity or embeddability on the torus.
The exponential factor 2^{n-8} suggests that the dominant contribution to subsemilattice growth comes from independent choices on a fixed-size core.

Load-bearing premise

The semilattice must be finite, because the proof enumerates and bounds the collection of all subsemilattices.

What would settle it

Exhibit a finite non-planar semilattice on n elements that possesses at least 127·2^{n-8} subsemilattices.

Figures

Figures reproduced from arXiv: 1906.12003 by G\'abor Cz\'edli.

**Figure 2.** Figure 2: These join-semilattices are planar Proof of Remark 1.3. For the sake of contradiction, suppose that L is a non-planar join-semilattice with at most seven elements. Then L ∪0 , see (4.7), is a non-planar lattice and |L ∪0 | ≤ 8. Since A0 is the only member of the Kelly–Rival list LKR with at most eight elements, it follows from (the Kelly–Rival) Theorem 3.1 that [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

read the original abstract

Let $L$ be a finite $n$-element semilattice. We prove that if $L$ has at least $127\cdot 2^{n-8}$ subsemilattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar semilattice with exactly $127\cdot 2^{n-8}-1$ subsemilattices.

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.

Desk Editor's Note private letter to a colleague

The paper pins down a sharp threshold of 127 * 2^{n-8} subsemilattices that forces planarity in finite n-element semilattices, with a matching non-planar example one below the bound.

read the letter

The central claim is that any finite semilattice on n elements with at least 127 * 2^{n-8} subsemilattices must be planar, and for n > 8 this is tight via an explicit non-planar construction with exactly one fewer subsemilattice. That specific constant and the sharpness example are the actual advance; prior bounds on subsemilattice counts were apparently looser, so this tightens the picture in a useful way for the subfield. The paper handles both directions in one package: the implication from the count to planarity, plus the counterexample showing the number cannot be lowered. The finiteness assumption is stated up front and is necessary for the counting argument, so there is no hidden gap there. The result is direct, with no fitted parameters or self-referential steps visible in the statement. The main soft spot is that the constant 127 almost certainly comes from exhaustive case analysis on small configurations or forbidden substructures, and without the full write-up it is impossible to check whether every case was covered or whether some overlap was missed. That kind of enumeration can hide small errors even when the overall logic is sound. The paper is aimed at specialists in finite semilattices, order theory, and planar posets. Someone classifying small semilattices or looking for numerical criteria to test planarity would get direct value from the bound and the example. It is narrow enough that it will not move broader mathematics, but within its niche the statement is precise and falsifiable. I would send it to peer review; the claim is concrete enough that referees can verify the proof steps and the construction without needing to guess at the authors' intent.

Referee Report

0 major / 1 minor

Summary. The paper proves that any finite n-element semilattice with at least 127·2^{n-8} subsemilattices is planar, and shows the bound is sharp for n>8 via an explicit non-planar construction achieving exactly one fewer subsemilattice.

Significance. If the proof and construction hold, this supplies a sharp extremal threshold connecting subsemilattice enumeration to Hasse-diagram planarity in finite semilattices. The matching lower-bound example is a clear strength, as is the direct (non-asymptotic) nature of the result.

minor comments (1)

The abstract and introduction should explicitly state the precise definition of 'planar semilattice' (Hasse diagram embeddable in the plane without crossings) to avoid any ambiguity for readers outside order theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. We appreciate the recognition of the direct, sharp nature of the extremal threshold relating subsemilattice counts to Hasse-diagram planarity.

Circularity Check

0 steps flagged

No significant circularity; direct proof and explicit construction

full rationale

The paper states a sharp extremal theorem for finite semilattices: a lower bound on the number of subsemilattices forces planarity of the Hasse diagram, with an explicit non-planar example achieving one fewer subsemilattice. The finiteness assumption is stated explicitly and is required for the counting argument. No equations, parameters, or claims reduce to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation consists of a combinatorial proof plus a concrete construction, both independent of the target statement.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a pure existence-and-implication proof in finite combinatorics on semilattices; it invokes only the standard definitions of semilattice, subsemilattice, and planarity together with basic counting arguments on finite sets.

pith-pipeline@v0.9.0 · 5587 in / 1037 out tokens · 23036 ms · 2026-05-25T14:08:31.690043+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · 5 internal anchors

[1]

K.: Yet two additional large numbers of subuniverses of ﬁnite lattices, Discussiones Mathematicae—General Algebra and Applications, accepted for publication

Ahmed, D., Horv´ ath, E. K.: Yet two additional large numbers of subuniverses of ﬁnite lattices, Discussiones Mathematicae—General Algebra and Applications, accepted for publication

work page
[2]

G.: On lattices whose ideals are all tolerance kernels

Chajda, I., Cz´ edli, G., Rosenberg, I. G.: On lattices whose ideals are all tolerance kernels. Acta Sci. Math. (Szeged) 61, 23–32 (1995)

work page 1995
[3]

Acta Universitatis Matthiae Belii, Series Mathematics Online, 22–28 (2018), http://actamath.savbb.sk/oacta2018003.shtml

Cz´ edli, G.: A note on ﬁnite lattices with many congruences. Acta Universitatis Matthiae Belii, Series Mathematics Online, 22–28 (2018), http://actamath.savbb.sk/oacta2018003.shtml

work page 2018
[4]

Lattices with many congruences are planar

Cz´ edli, G.: Lattices with many congruences are planar. Algebra Universalis 80:16 (2019) DOI: 10.1007/s00012-019-0589-1 http://arxiv.org/abs/1807.08384

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00012-019-0589-1 2019
[5]

Finite semilattices with many congruences

Cz´ edli, G.: Finite semilattices with many congruences. Order, published online July 7, 2018, DOI: 10.1007/s11083-018-9464-5 https://arxiv.org/abs/1801.01482

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s11083-018-9464-5 2018
[6]

Cz´ edli, G.: Eighty-three sublattices and planarity.https://arxiv.org/abs/1901.00572

work page internal anchor Pith review Pith/arXiv arXiv 1901
[7]

A note on lattices with many sublattices

Cz´ edli, G., Horv´ ath, E. K.: A note on lattices with many sublattices. http://arxiv.org/abs/1812.11512

work page internal anchor Pith review Pith/arXiv arXiv
[8]

Birkh¨ auser Verlag, Basel (2011) Math

Gr¨ atzer, G.: Lattice Theory: Foundation. Birkh¨ auser Verlag, Basel (2011) Math. (Szeged) 81, 25–32 (2015)

work page 2011
[9]

Kelly, D., Rival, I.: Planar lattices. Canad. J. Math. 27, 636–665 (1975

work page 1975
[10]

Kulin, J., Mure¸ san, C.: Some extremal values of the number of congruences of a ﬁnite lattice. https://arxiv.org/pdf/1801.05282 (2018) Cz´ edli: One hundred twenty-seven subsemilattices / Appendix 11 APPENDIX The rest of the paper is an appendix, which consists of the output ﬁle mentioned in the paragraph following Theorem 3.1. Version of the input file:...

work page internal anchor Pith review Pith/arXiv arXiv 2018

[1] [1]

K.: Yet two additional large numbers of subuniverses of ﬁnite lattices, Discussiones Mathematicae—General Algebra and Applications, accepted for publication

Ahmed, D., Horv´ ath, E. K.: Yet two additional large numbers of subuniverses of ﬁnite lattices, Discussiones Mathematicae—General Algebra and Applications, accepted for publication

work page

[2] [2]

G.: On lattices whose ideals are all tolerance kernels

Chajda, I., Cz´ edli, G., Rosenberg, I. G.: On lattices whose ideals are all tolerance kernels. Acta Sci. Math. (Szeged) 61, 23–32 (1995)

work page 1995

[3] [3]

Acta Universitatis Matthiae Belii, Series Mathematics Online, 22–28 (2018), http://actamath.savbb.sk/oacta2018003.shtml

Cz´ edli, G.: A note on ﬁnite lattices with many congruences. Acta Universitatis Matthiae Belii, Series Mathematics Online, 22–28 (2018), http://actamath.savbb.sk/oacta2018003.shtml

work page 2018

[4] [4]

Lattices with many congruences are planar

Cz´ edli, G.: Lattices with many congruences are planar. Algebra Universalis 80:16 (2019) DOI: 10.1007/s00012-019-0589-1 http://arxiv.org/abs/1807.08384

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00012-019-0589-1 2019

[5] [5]

Finite semilattices with many congruences

Cz´ edli, G.: Finite semilattices with many congruences. Order, published online July 7, 2018, DOI: 10.1007/s11083-018-9464-5 https://arxiv.org/abs/1801.01482

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s11083-018-9464-5 2018

[6] [6]

Cz´ edli, G.: Eighty-three sublattices and planarity.https://arxiv.org/abs/1901.00572

work page internal anchor Pith review Pith/arXiv arXiv 1901

[7] [7]

A note on lattices with many sublattices

Cz´ edli, G., Horv´ ath, E. K.: A note on lattices with many sublattices. http://arxiv.org/abs/1812.11512

work page internal anchor Pith review Pith/arXiv arXiv

[8] [8]

Birkh¨ auser Verlag, Basel (2011) Math

Gr¨ atzer, G.: Lattice Theory: Foundation. Birkh¨ auser Verlag, Basel (2011) Math. (Szeged) 81, 25–32 (2015)

work page 2011

[9] [9]

Kelly, D., Rival, I.: Planar lattices. Canad. J. Math. 27, 636–665 (1975

work page 1975

[10] [10]

Kulin, J., Mure¸ san, C.: Some extremal values of the number of congruences of a ﬁnite lattice. https://arxiv.org/pdf/1801.05282 (2018) Cz´ edli: One hundred twenty-seven subsemilattices / Appendix 11 APPENDIX The rest of the paper is an appendix, which consists of the output ﬁle mentioned in the paragraph following Theorem 3.1. Version of the input file:...

work page internal anchor Pith review Pith/arXiv arXiv 2018