The best extending cover-preserving geometric lattices of semimodular lattices

Peng He; Xue-Ping Wang

REVIEW 3 major objections 6 minor 9 references

The best extending cover-preserving geometric lattices of semimodular lattices

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper answers the 2010 question of the smallest geometric lattices extending a semimodular lattice, giving an algorithm that produces them all and proving each has exactly $|J(L)|$ atoms and length $\ell(L)$.

desk verdict Targets a real open problem and has a plausible algorithm, but Lemma 4.3's unproved Fig.8 assertion is false for N5; the classification is not established. read the letter →

arxiv 1908.00749 v2 pith:IR7SRGEW submitted 2019-08-02 math.CO

Peng He , Xue-ping Wang This is my paper · ORCID

classification math.CO MSC 06C1006B15

keywords finiteatomisticlatticesemimodulargeometriccover-preservingembeddingbestextendingjoin-irreducibleelementsatomsalgorithm

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

This paper addresses a question left open by the 2010 embedding theorem for semimodular lattices: given a finite semimodular lattice $L$, which geometric lattices $G$ contain $L$ as a cover-preserving sublattice and have the smallest possible size? The authors introduce a construction, Algorithm 3.1, that starts from any extending standard form of $L$ and outputs all geometric lattices in which $L$ sits cover-preservingly with the same length. Their main theorem states that every best extending cover-preserving geometric lattice of $L$ is the best output of this algorithm, and that each such lattice has exactly $|J(L)|$ atoms and length $\ell(L)$. This matters because it turns an existence result into a complete classification: the minimal geometric over-lattices of a finite semimodular lattice are finite in number and are governed by the join-irreducible elements of $L$.

What carries the argument

The machinery is the pair consisting of extending standard forms and Algorithm 3.1. Every finite lattice can be represented as a set system, a family of subsets of its atoms, and an extending standard form $P$ places new atoms below suitable non-atoms of $L$ so that $L$ becomes a cover-preserving sublattice of $P$ with $\ell(P)=\ell(L)$. The algorithm starts from the set-system copy $S_P$ of such a $P$ and repeatedly adds subsets $U$, and chains of subsets $W$, that pass three tests: $U$ is not contained in a previously present set of rank one less, $U$ does not contain the union of any independent set coming from the previous rank, and intersections with previous rank-one-less sets close inside $U$. These tests force the enlarged poset to remain closed under meets and to preserve joins inherited from $L$. The final condition (M), that adjoining any missing atom to a rank-$k$ element always raises the rank to $k+1$, guarantees semimodularity and hence that the output is geometric. Lemma 4.5 inverts the whole procedure, showing that any same-length geometric over-lattice must be obtainable in exactly this way.

What would settle it

Inspect finite semimodular lattices for a pair of incomparable elements $M,N$ with $M$ and $N$ both covering $M\cap N$ whose interval $[M\cap N,M\vee N]$ lacks the matched covering chains of Fig. 8; if such a pair exists, Lemma 4.3's cover-preserving conclusion can fail for a suitable extending standard form, and the claimed classification would not hold as stated.

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Extended reading notes

Core claim

The central claim is Theorem 5.1: every best extending cover-preserving geometric lattice of $L$ is the best one in the family $\mathcal{S}_{|J(L)|}$, where $\mathcal{S}_k$ consists of the outputs of Algorithm 3.1 with $k$ atoms. The proof has two parts. Lemma 4.5 shows that every finite geometric lattice $G$ with $L$ as a cover-preserving sublattice and $\ell(G)=\ell(L)$ is an output of Algorithm 3.1; Lemma 5.1 shows that any such output with more than $|J(L)|$ atoms can be strictly shrunk by deleting an atom while remaining a geometric lattice that still contains $L$ cover-preservingly. Together these imply that a geometric extension is smallest exactly when it has $|J(L)|$ atoms, and that all smallest extensions have length $\ell(L)$. Thus the paper's answer to the 2010 question is a full structural description rather than merely another existence proof.

Load-bearing premise

The proof of Lemma 4.3 assumes, without proof or citation, that any two incomparable elements of a finite semimodular lattice whose meet they both cover force the specific sublattice pictured in Fig. 8; this is what makes the joins computed in the algorithm's output agree with joins in $L$.

Editorial extensions

If this is right

Every best extending cover-preserving geometric lattice of a finite semimodular lattice has exactly $|J(L)|$ atoms and length $\ell(L)$, so the minimal size is determined by the join-irreducible elements of $L$.
Every finite geometric lattice that contains $L$ cover-preservingly and has the same length as $L$ is an output of Algorithm 3.1, so the algorithm lists the entire class of same-length geometric over-lattices.
Any same-length geometric extension with more than $|J(L)|$ atoms can be reduced by deleting one atom, so minimality is equivalent to having exactly $|J(L)|$ atoms.
The earlier existence result for cover-preserving embeddings into geometric lattices is sharpened from there being at least one such extension to a classification and construction of all minimal ones.

Reading between the lines

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

This suggests the minimal geometric over-lattices of $L$ are exactly the flat completions of the atomistic poset $S_P$; if so, enumeration algorithms for matroid flats could be adapted to list them for moderate $|J(L)|$.
Because different choices of $U$ and $W$ inside Algorithm 3.1 yield different outputs, the number of best extensions is a computable invariant of $L$; comparing this count among lattices with the same $|J(L)|$ and $\ell(L)$ might separate lattices that those two parameters cannot distinguish.
Condition (M) may be replaceable by a simpler exchange axiom, in which case the construction would translate directly into the language of single-element extensions of matroids, connecting the minimal over-lattice problem to matroid 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.

Desk Editor's Note

read the letter

Here is my read. The paper takes the Czédli-Schmidt open question seriously and makes a genuine attempt: it gives an algorithm (Algorithm 3.1) that is supposed to enumerate every cover-preserving geometric over-lattice of a finite semimodular lattice with the same length, and it extracts the consequences |A(G)|=|J(L)| and ℓ(G)=ℓ(L) for the minimal ones. The idea of shrinking a larger geometric over-lattice by deleting an inserted atom (Lemma 5.1) is appealing, and the worked examples make the intended construction concrete. That is real work and worth engaging with.

The soft spots are not cosmetic. Lemma 4.3 is the load-bearing step: it must show that every output Q of the algorithm actually contains L cover-preservingly. The proof asserts that if M and N are incomparable in the finite semimodular lattice T_L^P, then T_L^P contains the sublattice of Fig.8. No proof or citation is given. This is not a minor omission; the statement is false as a general semimodularity fact. In the pentagon N5, take the two incomparable elements M=a and N=b. Their join is 1, with ℓ(b)=1 and ℓ(1)=3. No chain b

There is a second, smaller gap: Lemma 4.5, the completeness direction, is a proof by repeated construction with phrases like "Repeating the process as above" and "Continuing as above". That may be fillable, but it is not a formal induction as written. Also Lemma 3.1 asserts that every extending standard form is an atomistic lattice; since E(L) is defined without a lattice requirement, that needs an argument too.

What does this add up to? The paper identifies a real open question and has a plausible strategy, and the examples check out. But the main theorem depends on a false structural assertion, so the classification is not proven as it stands. I would not cite the result in the next year. I would still send it to a serious referee, because the problem is important and a repaired version could be a good paper. The referee should be asked to focus on Lemma 4.3 first.

Bring it to reading group? Maybe, as a case study in where semimodularity arguments go wrong.

Referee Report

3 major / 6 minor

Summary. The paper studies the 2010 question of Czédli and Schmidt on the "best" cover-preserving embedding of a finite semimodular lattice L into a geometric lattice G. It introduces extending standard forms of L and an algorithm, Algorithm 3.1, intended to construct all finite geometric lattices Q that contain L as a cover-preserving sublattice with the same length. The main results are Lemma 4.5, which asserts that every finite geometric lattice Q with L as a cover-preserving sublattice and with ℓ(Q)=ℓ(L) is an output of the algorithm; Lemma 5.1, which asserts that every output with more than |J(L)| atoms can be strictly shrunk; and Theorem 5.1, which concludes that every best extending cover-preserving geometric lattice of L has exactly |J(L)| atoms and length ℓ(L). The paper also presents worked examples, in particular Examples 4.1 and 5.1, illustrating the algorithm and the claimed best lattices.

Significance. If the proofs were completed, the paper would resolve the Czédli-Schmidt question by providing a finite classification of the minimal geometric over-lattices and a constructive algorithm for obtaining them. The claimed equality |A(G)|=|J(L)| for every best extending geometric lattice is a clean and natural structural statement that would strengthen the existence theorem of Czédli and Schmidt. The manuscript is ambitious and contains instructive examples; the algorithm is described in full detail. However, the paper does not provide machine-checked proofs or reproducibility artifacts, and the central proof currently has gaps that are load-bearing for the classification. The significance is therefore conditional on completing the proof of Lemma 4.3 and formalizing the induction in Lemma 4.5.

major comments (3)

[§4, Lemma 4.3 (Fig. 8)] The proof of Lemma 4.3 contains the unproved assertion that if M and N are incomparable elements of the finite semimodular lattice T_L^P, then T_L^P contains the sublattice presented in Fig. 8. No proof or citation is supplied, and this is not a direct consequence of semimodularity alone: finite semimodular lattices are graded, but gradedness does not by itself produce the displayed cover-chain structure. The subsequent length identities ℓ(N_k)=t+k and ℓ(Z)=t+k+1 in Eq. (4) depend on that structure. The later step "there exists a subset ρ of A_SP(M) such that η∪ρ=π∈I_SP(N_k)" is also asserted without justification; S_P is only known to be atomistic, so Lemma 2.4, which is stated for geometric lattices, does not apply, and the extension of an independent set is not automatic. Since Lemma 4.3 is exactly what ensures that every output of Algorithm 3.1 has L as a cover-preserving sublattice, this gap is load-bearing for Lemmas 4.4, 4.5, 5.1, and Theorem 5.1.
[§4, Lemma 4.5 (pp. 19–21)] The completeness proof in Lemma 4.5 is compressed into "Repeating the process as above" and "Continuing as above." The induction must show that, at every stage, the newly added elements U (and U_1, W) satisfy conditions (i1)–(i3) or (j1)–(j3) with respect to the current set R_i, not merely with respect to Q; the manuscript verifies this explicitly only for the first addition from R_1=S_P. It also must prove that the sets R_i∪⋃{[0,M]_Q : ...} remain k-order normal subset lattices of Q. Without an explicit induction invariant, statements such as "R_3 is a 4 order normal subset lattice of Q" and the final conclusion Q=R_{m-1} are not established. This direction is essential because Lemma 4.5 is what converts the algorithm into a classification of all candidate geometric lattices.
[§4, Lemma 4.1 (Step A)] In Lemma 4.1, Step A, after proving ℓ_Q(E∩X)≤t−1 the text asserts: "there exists an element K∈ϕ_Q(t−1) such that E∩X⊆K, or E∩X∈A(Q)." This assertion needs an argument: in a poset that is not yet known to be graded, an element of length at most t−1 need not lie below an element of exact length t−1. If this step fails, the application of condition (i1) and the conclusion E∧_R M_{t−1}=E∩M_{t−1} do not follow. Since Lemma 4.1 is the proof that Algorithm 3.1 outputs a lattice, this issue is also load-bearing; the authors should either prove gradedness of the intermediate structures or revise the argument.

minor comments (6)

[Throughout] The notation for cover-preserving sublattice embedding, rendered as "֒→≺", is introduced inline; it would be clearer to define it in a displayed form or in the notation list.
[§4, Lemma 4.3] There is a typo: "a sublattice lattice" should read "a sublattice".
[§4, Lemma 4.4] The interval notation [∅, M] is used for [0, M], which is inconsistent with the bottom element 0 used elsewhere in the paper.
[§4, Lemma 4.5, Case 2] "a 2 order normal subset lattice of S" should read "of Q".
[References] Reference [5] contains a typo: "semimordular" should be "semimodular".
[Example 5.1] The statement that Q is the unique best extending cover-preserving geometric lattice "in the sense of isomorphism" is not proved; it should either be proved or labeled as an observation based on the algorithm run.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is a self-contained structural derivation, with the only concern being an unproved sublattice assertion in Lemma 4.3, which is a correctness gap rather than a circular reduction.

full rationale

The paper's derivation does not fit parameters, rename known results, or import its conclusion from self-citations. Theorem 1.1 from Czédli and Schmidt is used only as the external existence benchmark that fixes |A(G)|=|J(L)| and ℓ(G)=ℓ(L) for some over-lattice, and the subsequent lemmas then prove the reverse classification from first principles. Lemma 4.3 is intended to show every output Q of Algorithm 3.1 contains L as a cover-preserving sublattice, using formula (1), Lemma 4.2, and the semimodularity of T_L^P; the proof is structural and does not assume that Q has the desired covering property. Lemma 4.5 starts from an arbitrary geometric lattice Q with L cover-preserving and same length and reconstructs it as an output, which is a genuinely independent completeness direction. Lemma 5.1 shrinks any output with more than |J(L)| atoms to a smaller one, so the minimality conclusion in Theorem 5.1 is not assumed. The cited self-reference [7] is contextual only and not load-bearing. The one identifiable weakness is the unproved assertion in Lemma 4.3: "As T_L^P is a finite semimodular lattice, we know that T_L^P contains a sublattice lattice as presented in Fig.8," with no proof or citation supplied. This assertion is used to obtain the length identities for N_k and Z and the independent-set extension property needed to equate the joins in Q and T_L^P. If it fails, Lemma 4.3's conclusion is not established; however, failing to prove a structural claim is a mathematical gap, not a circular derivation. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central claim depends on standard lattice-theoretic results (Crawley-Dilworth, Czédli-Schmidt), on the domain assumption that intervals of geometric lattices are geometric, and on two paper-specific structural claims: the unproved Fig.8 sublattice in Lemma 4.3 and the completeness induction in Lemma 4.5. No free parameters are fitted. The 'invented entities' are new mathematical definitions, not physically postulated objects.

assumptions (5)

standard math Birkhoff's condition is equivalent to semimodularity for strongly atomic algebraic lattices (Crawley-Dilworth Theorem 3.7).
Invoked in Section 1 to characterize semimodularity and used throughout the proof framework.
standard math Independent sets in a geometric lattice satisfy the characterization of Crawley-Dilworth Theorem 6.5, used to define and reason about the independent functions I_P(x).
This is the basis of Lemma 2.4 and the independent-set machinery behind Algorithm 3.1.
domain assumption Czédli-Schmidt Theorem 1.1 guarantees the existence of an extending standard form P of L with |A(P)|=|J(L)| and ℓ(P)=ℓ(L).
The algorithm starts from an extending standard form P; the paper needs at least one such P to exist, and the cited theorem (or Gratzer-Kiss) provides it. This also ensures S_{|J(L)|} is nonempty.
domain assumption Every interval [x,y] of a geometric lattice is geometric and ranked, so any cover-preserving image of L in G can be restricted to an interval of length ℓ(L).
Used in the paragraph before Theorem 5.1 to reduce the search to geometric lattices of the same length as L.
ad hoc to paper If M and N are incomparable elements of a finite semimodular lattice and M∨N covers M and N, then the lattice contains the sublattice in Fig.8.
This structural fact is asserted without proof in Lemma 4.3 and is load-bearing for showing the join in the constructed lattice Q agrees with the join in L.

invented entities (2)

Extending standard form of L
purpose: A constructed lattice P obtained by inserting new atoms below each non-atomic join-irreducible element of L, used as the starting point of Algorithm 3.1.
This is a mathematical construction defined in Section 3; its validity is judged internally by the proofs, and it has no falsifiable external handle.
Atomistic partially ordered set (Definition 2.2)
purpose: A poset with a strengthened atomistic property used to represent all candidates as families of subsets of atoms.
A new definition introduced to support the set-theoretic representation in Lemma 2.3; it is a mathematical object with no external empirical content.

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Cite this review

Pith. "Pith review of The best extending cover-preserving geometric lattices of semimodular lattices." pith.science (2026). https://pith.science/paper/IR7SRGEW

@misc{pith2026190800749,
  author       = {Pith},
  title        = {Pith review of: The best extending cover-preserving geometric lattices of semimodular lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IR7SRGEW}},
  note         = {Machine review of arXiv:1908.00749}
}

abstract

In 2010, G\'{a}bor Cz\'{e}dli and E. Tam\'{a}s Schmidt mentioned that the best cover-preserving embedding of a given semimodular lattice is not known yet [A cover-preserving embedding of semimodular lattices into geometric lattices, Advances in Mathematics 225 (2010) 2455-2463]. That is to say: What are the geometric lattices $G$ such that a given finite semimodular lattice $L$ has a cover-preserving embedding into $G$ with the smallest $|G|$? In this paper, we propose an algorithm to calculate all the best extending cover-preserving geometric lattices $G$ of a given semimodular lattice $L$ and prove that the length and the number of atoms of every best extending cover-preserving geometric lattice $G$ equal the length of $L$ and the number of non-zero join-irreducible elements of $L$, respectively. Therefore, we comprehend the best cover-preserving embedding of a given semimodular lattice.

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Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages

[1]

Birkhoﬀ, Lattice Theory, vol.XXV, 3rd ed., American Mathemat ical Society Col- loquium Publications, Providence, RI, 1973

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P. Pudl´ ak, J. T¨ uma, Every ﬁnite lattice can be embedded in a par tition lattice, Algebra Universalis 10 (1980) 74-95

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[9]

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Pith tools

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

[1] [1]

Birkhoﬀ, Lattice Theory, vol.XXV, 3rd ed., American Mathemat ical Society Col- loquium Publications, Providence, RI, 1973

G. Birkhoﬀ, Lattice Theory, vol.XXV, 3rd ed., American Mathemat ical Society Col- loquium Publications, Providence, RI, 1973

1973

[2] [2]

J. R. B¨ uchi, Representation of complete lattices by sets, Port ugaliae mathematica 11 (1952) 151-167. 26

work page 1952

[3] [3]

Crawley, R

P. Crawley, R. P. Dilworth, Algebraic Theory of Lattices, Prentic e Hall, Englewood Cliﬀs, NJ, 1973

work page 1973

[4] [4]

Cz´ edli, E

G. Cz´ edli, E. T. Schmidt, A cover-preserving embedding of semim odular lattices into geometric lattices, Advances in Mathematics 225 (2010) 2455-246 3

work page 2010

[5] [5]

Gr¨ atzer, E

G. Gr¨ atzer, E. W. Kiss, A construction of semimordular lattices , Order 2 (1986) 351-365

work page 1986

[6] [6]

Gr¨ atzer, F

G. Gr¨ atzer, F. Wehrung, Lattice Theory: Special Topics and A pplications, Birkh¨ aser Verlag, Basel, 2014

work page 2014

[7] [7]

Peng He, Xue-ping Wang, A characterization of a semimodular latt ice, Studia Logica 106 (2018) 691-698

work page 2018

[8] [8]

Pudl´ ak, J

P. Pudl´ ak, J. T¨ uma, Every ﬁnite lattice can be embedded in a par tition lattice, Algebra Universalis 10 (1980) 74-95

work page 1980