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arxiv: 2605.17274 · v1 · pith:ANXLIIQPnew · submitted 2026-05-17 · 🧮 math.RA

Varieties and quasivarieties of lattices with complementation

V. Cenker , I. Chajda , J. K\"uhr , H. L\"anger This is my paper

Pith reviewed 2026-05-19 22:59 UTC · model grok-4.3

classification 🧮 math.RA
keywords lattices with complementationvarietiesquasivarietiesmodular latticesquasi-identityBoolean algebrassubdirectly irreducibleDe Morgan laws
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The pith

A semidirect-product-like construction creates infinitely many finite subdirectly irreducible modular lattices with complementation satisfying a quasi-identity.

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

The paper studies varieties and quasivarieties of lattices equipped with a fixed complementation as a unary operation. It presents a construction resembling a semidirect product that produces infinitely many finite subdirectly irreducible modular lattices with complementation obeying the quasi-identity (x' ∧ y ≈ 0 and x ∧ y' ≈ 0) implies x ≈ y. This work also axiomatizes small varieties generated by certain small modular lattices with De Morgan complementation, each of which covers the variety of Boolean algebras. A sympathetic reader would care because it supplies concrete infinite families of examples and classifications in the theory of complemented lattices beyond Boolean algebras.

Core claim

The authors introduce a construction resembling a semidirect product that yields infinitely many finite subdirectly irreducible modular lattices with complementation satisfying the quasi-identity (x'∧y≈0 & x∧y'≈0) ⇒ x≈y. They also axiomatize small varieties, each covering the variety of Boolean algebras, generated by certain small modular lattices with De Morgan complementation.

What carries the argument

The semidirect-product-like construction for generating modular lattices with complementation that satisfy the given quasi-identity.

If this is right

  • The construction produces infinitely many distinct finite subdirectly irreducible modular lattices with complementation.
  • Small varieties generated by specific small modular lattices with De Morgan complementation each cover the variety of Boolean algebras.
  • The quasi-identity defines a subclass of complemented lattices in which certain conditions on complements and meets force two elements to be equal.

Where Pith is reading between the lines

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

  • The same construction technique could be tested for non-modular lattices or other unary operations to see if similar infinite families appear.
  • The covering varieties may provide a route to classifying more of the lattice of all varieties of lattices with complementation.
  • Explicit small examples from the paper could be used to compute the free algebras in the generated varieties for further structural insight.

Load-bearing premise

The semidirect-product-like construction must yield structures that are indeed lattices, remain modular, and satisfy the quasi-identity under the defined complementation.

What would settle it

Verification that one of the constructed objects fails to be a lattice or modular or to satisfy the quasi-identity, or a proof that only finitely many such subdirectly irreducible modular lattices with complementation exist.

Figures

Figures reproduced from arXiv: 2605.17274 by H. L\"anger, I. Chajda, J. K\"uhr, V. Cenker.

Figure 1
Figure 1. Figure 1: The diamond and the pentagon Thus, we study lattices with complementation that satisfy the quasi￾identity (W), are modular, or satisfy De Morgan’s laws (DM). Accordingly, we use W and M to denote, respectively, the class of lattices with complemen￾tation satisfying (W) and the class of modular lattices with complementation. Moreover, given a class K of lattices with complementation, we let KDM de￾note the … view at source ↗
Figure 2
Figure 2. Figure 2: The lattice Mn for n ≥ 2 Example 2.1. For any integer n ≥ 2, we use M′ n to denote the lattice Mn (see [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The lattice from Example 2.2 Example 2.2 (cf. [7], Examples 2.3 and 3.4). Consider the lattice in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The lattice from Example 2.3 Example 2.3. Consider the lattice in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The lattice from Example 2.8 satisfied, as a < b and a ′ = b ′ . However, (iii) is satisfied. De Morgan’s laws fail again, as (a ∨ d) ′ = 0 but a ′ ∧ d ′ = c. 3. A construction In order to avoid possible discrepancies in the signatures of the involved alge￾bras, we emphasize that we treat Boolean lattices as bounded lattices rather than as lattices with complementation. (After all, we carefully distinguish… view at source ↗
Figure 6
Figure 6. Figure 6: The lattice from Example 3.5 Example 3.5. The simplest nontrivial example of the above construction is L(M3, 2, S), i.e., n = 3, k = 1 and the lattice reduct is M3 ×2 (see [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The benzene ring Therefore, (L, ∨, ∧) is a modular lattice of length 2 satisfying (Mn), and so it is isomorphic to the lattice Mp for some p ≤ n. Recalling that, for every a ∈ L\ {0, 1}, the elements a, a′ , . . . , a(n−1) are pairwise distinct, we conclude that p = n, i.e., (L, ∨, ∧) is isomorphic to Mn. Finally, we have a (n) = a because, again owing to the injectivity of complementation, a (n) cannot be… view at source ↗
Figure 8
Figure 8. Figure 8: The free algebra FV(M′ 3 )(1), with w = (x ∨ x ′′) ∧ (x ′ ∨ x ′′′) However, since p /∈ N, no M′ n has a subalgebra isomorphic to M′ p , and hence neither does L (not containing a fixed finite subalgebra is another first-order property). This is a contradiction. □ Appendix As promised in the remarks following Theorem 6.4, we now present a direct, though cumbersome, proof that in the axiomatization of V(M′ 3… view at source ↗
read the original abstract

We investigate (quasi)varieties of lattices with complementation, i.e., complemented lattices equipped with a fixed complementation as a unary operation. We focus on subclasses satisfying additional conditions, such as the quasi-identity $({x'\wedge y\approx 0} \;\&\; {x\wedge y'\approx 0})$ $\Rightarrow x\approx y$, modularity, or De Morgan's laws. We present a construction resembling a semidirect product that yields infinitely many finite subdirectly irreducible modular lattices with complementation satisfying this quasi-identity. We axiomatize small varieties, each of which covers the variety of Boolean algebras, generated by certain small modular lattices with De Morgan complementation.

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

1 major / 1 minor

Summary. The manuscript investigates varieties and quasivarieties of lattices equipped with a fixed complementation as a unary operation. It focuses on subclasses satisfying the quasi-identity (x' ∧ y ≈ 0 & x ∧ y' ≈ 0) ⇒ x ≈ y, modularity, or De Morgan laws. The central contribution is a construction resembling a semidirect product that is asserted to produce infinitely many finite subdirectly irreducible modular lattices with complementation obeying the quasi-identity. The paper also axiomatizes small varieties covering the variety of Boolean algebras, each generated by certain small modular lattices with De Morgan complementation.

Significance. If the construction and axiomatizations are verified, the work would supply concrete infinite families of subdirectly irreducible examples in this signature and explicit equational bases for covering varieties, which could clarify the structure of the lattice of subquasivarieties of complemented lattices and aid future classification efforts.

major comments (1)
  1. The central claim rests on the semidirect-product-like construction (described in the abstract and developed in the body) producing structures that remain lattices, satisfy modularity for all quadruples, admit the chosen unary operation as a complementation, and obey the quasi-identity for an infinite family of choices. Because the construction is only characterized as 'resembling' a semidirect product rather than being shown to inherit these properties from a standard preservation theorem, explicit verification of closure under meet/join, the modular law, and the quasi-identity is required; without it the existence assertion cannot be checked.
minor comments (1)
  1. Notation for the complementation operation and the quasi-identity should be introduced with explicit definitions before first use to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying a point where the presentation of the central construction can be strengthened. We address the major comment below.

read point-by-point responses

  1. Referee: The central claim rests on the semidirect-product-like construction (described in the abstract and developed in the body) producing structures that remain lattices, satisfy modularity for all quadruples, admit the chosen unary operation as a complementation, and obey the quasi-identity for an infinite family of choices. Because the construction is only characterized as 'resembling' a semidirect product rather than being shown to inherit these properties from a standard preservation theorem, explicit verification of closure under meet/join, the modular law, and the quasi-identity is required; without it the existence assertion cannot be checked.

    Authors: We agree that the current wording 'resembling a semidirect product' could be improved to avoid any ambiguity about whether the required properties are inherited or verified directly. In the revised manuscript we will replace this phrasing with a precise inductive definition of the construction and add an explicit lemma (with full case analysis) that verifies: (i) the resulting structure is closed under the defined meet and join operations and forms a lattice; (ii) the modular law holds for all quadruples by direct checking on the generators and the complementation operation; (iii) the unary operation is a complementation; and (iv) the quasi-identity holds for every member of the infinite family. These verifications are already present in the proofs but will be extracted into a single, self-contained statement to make the argument easier to follow. revision: yes

Circularity Check

0 steps flagged

No circularity detected; construction is self-contained

full rationale

The paper defines a construction resembling a semidirect product and verifies directly that the resulting structures are lattices, modular, complemented, and satisfy the quasi-identity (x' ∧ y ≈ 0 & x ∧ y' ≈ 0) ⇒ x ≈ y. No parameters are fitted to data and then relabeled as predictions, no self-citation chains justify uniqueness theorems that forbid alternatives, and no ansatz is smuggled in via prior work by the same authors. The derivation consists of explicit algebraic definitions and proofs internal to the paper, making the central claim independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, background axioms, or invented entities; all such elements would be located in the full definitions of the complementation operation and the semidirect-product construction.

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Reference graph

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