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arxiv: 2507.22682 · v2 · submitted 2025-07-30 · 🧮 math.RA

Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond

K. Adaricheva , A. Mata , S. Silberger , A. Zamojska-Dzienio This is my paper

Pith reviewed 2026-05-19 02:59 UTC · model grok-4.3

classification 🧮 math.RA
keywords semidistributive latticesmaximal sublatticeslattice complementsintervalsconvex geometriesjoin-semidistributive latticesmeet-semidistributive lattices
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The pith

Complements of maximal sublattices in finite semidistributive lattices are intervals.

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

The paper studies maximal sublattices of finite semidistributive lattices by examining their complements in the larger lattice. It centers on the conjecture that these complements are always intervals, a fact already established for the subclass of bounded lattices. The authors also analyze the same question inside the broader join-semidistributive and meet-semidistributive classes. For the special case of convex geometries of convex dimension 2, they supply an explicit description of every such complement together with a systematic way to locate them all.

Core claim

We conjecture that the complement of every maximal sublattice of a finite semidistributive lattice is an interval. The same property is already known for bounded lattices. We further examine the complements inside join-semidistributive lattices and inside meet-semidistributive lattices. In the subclass consisting of convex geometries of convex dimension 2 we give a complete description of the complements and a procedure for finding all of them.

What carries the argument

The complement of a maximal sublattice, conjectured to coincide with an interval inside the ambient finite semidistributive lattice.

If this is right

  • If the conjecture holds, the position of every maximal sublattice inside a finite semidistributive lattice is completely determined by an interval.
  • For convex geometries of convex dimension 2 the full list of maximal-sublattice complements can be generated by a concrete procedure.
  • The same structural description supplies an explicit method for enumerating all maximal sublattices in that subclass.

Where Pith is reading between the lines

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

  • Analogous interval-complement statements might be tested inside other natural classes of finite lattices.
  • Systematic computer searches over small semidistributive lattices could supply supporting evidence or locate a counter-example.
  • The results may clarify how maximal sublattices interact with congruence lattices or with representation theorems for semidistributive structures.

Load-bearing premise

The lattices under study must be finite and semidistributive (or belong to one of its join- or meet-semidistributive subclasses) for the stated conjecture and description to apply.

What would settle it

Any finite semidistributive lattice that contains a maximal sublattice whose complement is not an interval would disprove the conjecture.

Figures

Figures reproduced from arXiv: 2507.22682 by A. Mata, A. Zamojska-Dzienio, K. Adaricheva, S. Silberger.

Figure 1
Figure 1. Figure 1: The inclusion relations between classes of lattices we focus on in this paper. The class of distributive lattices is contained in the class of bounded lattices; by which we mean the images of bounded homomorphic images of free lattices (see the precise definition in Section 2). The complements of maximal sublattices of bounded lattices were also shown to be intervals in [2, Theorem 7]. This suggests the fo… view at source ↗
Figure 2
Figure 2. Figure 2: The figure above shows that in a CG with cdim = n, which will also be SD∨, there can be a complement of a maximal sublattice with n distinct maximal elements. Here {2, 12, 24, 25, 26, . . . , 2n} is the complement of a maximal sublattice with all but 2 maximal elements. A convex geometry (CG) ⟨G, ∨, ∧⟩ is an SD∨ lattice which is also lower semi-modular: ∀ x, y ∈ G x ≺ x ∨ y =⇒ x ∧ y ≺ y, where a ≺ b means … view at source ↗
Figure 3
Figure 3. Figure 3: This lattice is an example of a lattice that is neither SD∨ nor SD∧ that has a complement of a maximal sublattice with more than one minimum and more than one maximum. The set {w, x, y, z} is the complement of a maximal sublattice. It is not SD∨ because y = u ∨ x = u ∨ z, but u ∨ (x ∧ z) = u ∨ 0 = u ̸= u ∨ x. Similarly, it is not SD∧ because x = v ∧ w = v ∧ y, but v ∧ (w ∨ y) = v ∧ 1 = v ̸= v ∧ w. As we ca… view at source ↗
Figure 4
Figure 4. Figure 4: The lattice L on the right is a distributive lattice the lattice L ′ to the right is an example of a lattice derived by doubling the interval [a, b]. Day’s 1992 paper [9] shows that bounded lattices are precisely those obtained by a sequence of interval doublings of this nature starting with a distributive lattice. Thus, L ′ is bounded, but you can check that the interval [a, c] is the complement of a maxi… view at source ↗
Figure 5
Figure 5. Figure 5: This lattice is an example of an SD lattice that is not bounded discovered by Jonsson and Nation [11, [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: This graphic is to illustrate Lemma 4.3. This lattice happens to be a CG with cdim = 4 and hence is SD∨. We see that d1 = a 1 1 ∧ a 2 1 , d2 = a 1 2 ∧ a 2 2 , and y = a 1 1 ∨ a 1 2 = a 1 1∨a 2 2 = a 2 1∨a 1 2 = a 2 1∨a 2 2 . Thus, Lemma 4.3 then guarantees what is shown, that y = d1∨d2. Let L be a lattice. Then L is SD∨ if and only if it satisfies the following condition w = _ i ai = _ j bj implies w = _ i… view at source ↗
Figure 7
Figure 7. Figure 7: This figure illustrates Corollary 5.3. In the figure, there is a dotted arrow from point a to point b if a is a cj or a cm of b, and the corollary says that if the complement of a maximal sublattice has such a “spiraling out” chain, then it is an interval [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: This figure illustrates the situation that Lemma 5.4 does not allow. The dotted arrow from u1 to x indicates u1 is a scm of x and the arrow from x to u2 indicates x is a scj of u2. Symmetrically, assume that m1 ∧ m2 is in [x, y]. Then x ≤ m1, m2. Also, w = m1 ∧ m2 ∧ b ′ . Therefore, y ≥ m1 or y ≥ m2, which would imply m1 ∈ [x, y] or m2 ∈ [x, y], a contradiction. Therefore, L\[x, y] must be a sublattice. □ … view at source ↗
Figure 9
Figure 9. Figure 9: These images illustrate the proof of Theorem 5.5. As before, there is a dotted arrow from point a to point b if either a is a cj or a cm of b. Proof. Suppose x = u1 ∧ b, where b is the meet of all of the other meetands of x. Take s = u2 ∧ b. Then x ≤ s ≤ u2. Note that since u1 is a scm of x and u2 ̸≤ u1, we get x < s. If s ∈ C, then either s ≥ t or s ≤ t. If s ≥ t, then b ≥ t and so u1 ∧ b ≥ t > x, a contr… view at source ↗
Figure 10
Figure 10. Figure 10: The three types of complements of complements of maximal sublattices of CGs with cdim = 2, explained in Example 6.2. (5) For any x ∈ X, [∅, Ci(x)] = [(x), Ci(x)]∪˙ [∅, Ci(p)], where Ci(p) ≺ Ci(x). (6) Suppose Ci(x) = X for some x ∈ X. Then (x) ∈ Ci ′ . Proof. Fix a convex geometry G with cdim = 2, generating chains C1 and C2, and base set X. 1. Fix x ∈ X. We know that (x) = m1 ∧ m2 for some meet irreducib… view at source ↗
read the original abstract

We study maximal sublattices of finite semidistributive lattices via their complements. We focus on the conjecture that such complements are always intervals, which is known to be true for bounded lattices. Since the class of semidistributive lattices is the intersection of classes of join- and meet-semidistributive lattices, we study also complements for these classes, and in particular convex geometries of convex dimension 2, which is a subclass of join-semidistributive lattices. In the latter case, we describe the complements of maximal sublattices completely, as well as the procedure of finding all complements of maximal sublattices.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript proposes a conjecture that the complements of maximal sublattices in finite semidistributive lattices are always intervals, a property known to hold for bounded lattices. It examines the related classes of join- and meet-semidistributive lattices and supplies a complete description of such complements together with an enumeration procedure for the subclass of convex geometries of convex dimension 2.

Significance. If the conjecture is confirmed it would supply a useful structural fact about maximal sublattices in semidistributive lattices. The explicit description and enumeration procedure for convex geometries of convex dimension 2 constitute a concrete advance that can be applied immediately in the study of join-semidistributive lattices and convex geometries.

minor comments (2)
  1. The introduction should include a short reference or citation for the known result on bounded lattices to help readers locate the background material.
  2. A few concrete examples illustrating the enumeration procedure for convex geometries of dimension 2 would improve readability and allow readers to verify the description.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the conjecture, and recommendation for minor revision. The explicit description and enumeration procedure for convex geometries of convex dimension 2 are indeed presented as a concrete contribution in the manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript states a conjecture that complements of maximal sublattices are intervals in finite semidistributive lattices (extending a known result for the bounded case) and supplies an explicit description plus enumeration procedure for the subclass of convex geometries of convex dimension 2. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the argument rests on standard definitions of semidistributive lattices and convex geometries without renaming known results or smuggling ansatzes. The derivation chain is self-contained against external benchmarks in lattice theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background definitions and properties of lattices, semidistributivity, and convex geometries without introducing new free parameters or postulated entities.

axioms (2)
  • domain assumption Finite semidistributive lattices satisfy the standard join- and meet-semidistributive identities.
    Invoked throughout the study of complements of maximal sublattices.
  • domain assumption Convex geometries of convex dimension 2 form a subclass of join-semidistributive lattices.
    Used to obtain the complete description of complements.

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Reference graph

Works this paper leans on

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