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arxiv: 2509.04417 · v2 · submitted 2025-09-04 · 🧮 math.RA

Dual spaces of lattices and semidistributive lattices

Andrew Craig , Miroslav Haviar , Jos\'e S\~ao Jo\~ao This is my paper

Pith reviewed 2026-05-18 19:29 UTC · model grok-4.3

classification 🧮 math.RA MSC 06B0506B23
keywords lattice dual spacesPloščica spacessemidistributive latticesrepresentation theoremsdigraph relationsUrquhart L-spaces
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The pith

Every Ploščica space is the dual space of some lattice.

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

The authors translate Urquhart's representation of general lattices, which uses two quasi-orders on a space, into a version that uses a single digraph relation instead. They prove that any space built this way arises exactly as the dual of some lattice. The same setup lets them begin describing infinite join and meet semidistributive lattices by the properties of their dual digraphs, building on earlier finite results.

Core claim

Every Ploščica space is the dual space of some general lattice. The digraph relation is chosen so that it captures the lattice operations in the same way Urquhart's two quasi-orders do.

What carries the argument

Ploščica space: a compact totally order-disconnected topological space equipped with a digraph relation that replaces the pair of quasi-orders in Urquhart's L-spaces.

If this is right

  • Any lattice, finite or infinite, admits a dual representation via a digraph on a topological space.
  • Properties of join and meet semidistributive lattices can be read from the structure of their dual digraphs.
  • Examples show how the dual digraphs behave for concrete lattices and for semidistributive cases.
  • Three open problems point to further questions about these representations.

Where Pith is reading between the lines

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

  • The representation may make it easier to prove lattice identities by moving to the dual digraph setting.
  • Similar translations could apply to other classes of lattices beyond the semidistributive ones.
  • Checking the dual digraphs of known infinite semidistributive lattices would test how far the finite characterizations extend.

Load-bearing premise

The chosen digraph relation must encode joins and meets of the lattice exactly when it is built from the two-quasi-order definition.

What would settle it

A concrete Ploščica space whose digraph fails to match the prime filters or ideals of any lattice would disprove the claim.

Figures

Figures reproduced from arXiv: 2509.04417 by Andrew Craig, Jos\'e S\~ao Jo\~ao, Miroslav Haviar.

Figure 1
Figure 1. Figure 1: The six lattices L, which cannot appear as sub￾lattices of semidistributive lattices, and their duals XL In case the lattice L is finite, every filter is the up-set of a unique ele￾ment and every ideal is the down-set of a unique element. Hence in a finite lattice L we can represent every disjoint filter-ideal pair ⟨F, I⟩ by an ordered pair ⟨↑x, ↓y⟩ where x = V F and y = W I. Thus for finite lattices we ha… view at source ↗
Figure 2
Figure 2. Figure 2: Illustrating the proof of Theorem 4.6 The following proposition is essential for our characterisation in Theo￾rem 4.14. The original result is [15, Lemma 2.1], although we remark that the formulation below taken from [1, Theorem 3-1.27] is not the same as the statement which appears in [15, Lemma 2.1]. For the labels of the lattices below we refer to [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The eight sublattices obtained in each case of Proposition 4.13. Case 5: Only (A) holds. Lemma 4.12(ii), (A) and the fact that (B) does not hold give us that I ∧ G = F ∧ G < I ∧ F. Since (C) does not hold, and using Lemma 4.11, we get F∨G < I∨G = I∨F. We get a sublattice isomorphic to L3. Case 6: Only (B) holds. Lemma 4.12(ii), (B) and the fact that (A) does not hold give us that I ∧ F = F ∧ G < I ∧ G. Sin… view at source ↗
Figure 4
Figure 4. Figure 4: The infinite semidistributive lattice Oω, and the core of its dual space (Example 5.1) for all i, j, k ∈ ω with i < j. Swapping all a’s with b’s above will give us, with the previous sublist, the list of all MDFIPs in the edge relation E. The lattice and it dual space are drawn in [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: ) This lattice is not join semidistributive since [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: The infinite meet but not join semidistributive lattice Oˆ ω, and the core of its dual space (Example 5.2) where Ia = {0, a0, a1, a2, ...} and Ib = {0, b0, b1, b2, ...}. One can check that MDFIPs, in which a’s appear, naturally arising in the edge relation E are given by aj+1ajEai+1ai , ai+1aiEbk+1bk, aj+1ajEa0y, b0aEaωIa, b0aEai+1ai , ai+1aiEb0a, ai+1aiEbωIb, aωIaEbk+1bk, aωIaEbωIb, acEbc, acEaj+1aj , acE… view at source ↗
Figure 6
Figure 6. Figure 6: The infinite lattice R, which is neither join semidistributive nor meet semidistributive, and its dual space (Example 5.3) Example 5.3 (The “rocket”, an infinite lattice that is neither meet semidis￾tributive nor join semidistributive). Let R be the lattice on left in [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
read the original abstract

Birkhoff's 1937 dual representation of finite distributive lattices via finite posets was in 1970 extended to a dual representation of arbitrary distributive lattices via compact totally order-disconnected topological spaces by Priestley. This result enabled the development of natural duality theory in the 1980s by Davey and Werner, later on also in collaboration with Clark and Priestley. In 1978 Urquhart extended Priestley's representation to general lattices via compact doubly quasi-ordered topological spaces (L-spaces). In 1995 Plo\v{s}\v{c}ica presented Urquhart's representation in the spirit of natural duality theory by replacing, on the dual side, Urquhart's two quasiorders with a digraph relation generalising Priestley's order relation. In this paper we translate, following the spirit of natural duality theory, Urquhart's L-spaces into newly introduced \emph{Plo\v{s}\v{c}ica spaces}. We then prove that every Plo\v{s}\v{c}ica space is the dual space of some general lattice. Based on the authors' 2022 characterisation of finite join and meet semidistributive lattices via their dual digraphs, we initiate a study of general (possibly infinite) join and meet semidistributive lattices via their dual digraphs. We illustrate our results on examples and formulate three open problems.

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 / 2 minor

Summary. The paper translates Urquhart's L-spaces into newly defined Ploščica spaces by replacing the pair of quasi-orders with a single digraph relation. It proves that every Ploščica space is the dual space of some general lattice and initiates the study of (possibly infinite) join and meet semidistributive lattices via their dual digraphs, extending the authors' 2022 finite characterization. Illustrative examples are given and three open problems are formulated.

Significance. If the representation theorem holds, the work supplies a natural-duality-style framework that unifies the treatment of general lattices and opens a route to infinite semidistributive lattices via digraph duals. Strengths include the independent proofs for the Ploščica representation and the axiomatic setup for the semidistributive extension; these are parameter-free and supply falsifiable predictions through the stated open problems.

major comments (1)
  1. [Introduction and proof of the representation theorem] Introduction and proof of the representation theorem: the central claim that every Ploščica space is the dual of some lattice requires that the single digraph relation preserves the separation and interpolation axioms of Urquhart's two-quasi-order L-spaces. No explicit verification that the Ploščica axioms imply the original L-space axioms is supplied; without it the recovery of associative lattice operations and exact dual-space correspondence remains unconfirmed for the infinite case.
minor comments (2)
  1. [Abstract] The reference to Ploščica (1995) in the abstract and introduction should be expanded to full bibliographic form.
  2. [Examples] The examples section would be strengthened by an explicit computation of the dual digraph for at least one infinite semidistributive lattice.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting an important point about the representation theorem. We address the major comment below and will revise the paper accordingly to improve clarity.

read point-by-point responses

  1. Referee: Introduction and proof of the representation theorem: the central claim that every Ploščica space is the dual of some lattice requires that the single digraph relation preserves the separation and interpolation axioms of Urquhart's two-quasi-order L-spaces. No explicit verification that the Ploščica axioms imply the original L-space axioms is supplied; without it the recovery of associative lattice operations and exact dual-space correspondence remains unconfirmed for the infinite case.

    Authors: We thank the referee for this observation. The Ploščica axioms in Definition 3.2 are designed so that the induced pair of quasi-orders on the space satisfy Urquhart's separation and interpolation conditions by construction; the representation proof in Section 4 then recovers the lattice operations directly from the digraph and verifies associativity and the dual-space correspondence using these axioms. The argument applies verbatim to the infinite case. To make the implication fully explicit, we will insert a short lemma (new Lemma 3.5) that derives the two L-space axioms from the four Ploščica axioms. This addition will not change any results but will render the correspondence transparent. revision: yes

Circularity Check

0 steps flagged

Representation theorem for Ploščica spaces relies on independent definitions and proofs

full rationale

The paper introduces Ploščica spaces via a direct translation of Urquhart's L-spaces that replaces the pair of quasi-orders with a single digraph relation, then supplies an explicit construction proving every such space is the dual of a lattice. This chain does not reduce any central claim to a self-definitional equation, a fitted parameter renamed as prediction, or a load-bearing self-citation whose content is unverified. The 2022 citation is invoked only for the separate finite semidistributive characterization and does not underpin the general representation result, which remains self-contained against the external Priestley and Urquhart benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The claims rest on standard background theorems of lattice theory and topology together with the newly introduced definitions of Ploščica spaces and dual digraphs; no free parameters or data-fitting are present.

axioms (1)
  • standard math Background results from lattice theory, topology, and natural duality theory (Priestley, Urquhart, Ploščica 1995).
    The paper explicitly builds upon these established representation theorems.
invented entities (2)
  • Ploščica space no independent evidence
    purpose: Dual representation of arbitrary lattices via a single digraph relation.
    Newly defined in the paper as a reformulation of Urquhart's L-spaces.
  • Dual digraph for infinite semidistributive lattices no independent evidence
    purpose: Characterization of join and meet semidistributive lattices beyond the finite case.
    Extension of the authors' 2022 finite-case result.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    and Nation, J.B.: Classes of semidistributive lattices

    Adaricheva, K. and Nation, J.B.: Classes of semidistributive lattices. In: Gr¨ atzer G., Wehrung F. (eds). Lattice Theory: Special Topics and Applica- tions. Birkh¨ auser, pp. 59–101 (2016)

    work page 2016

  2. [2]

    Birkhoff, G.: Rings of sets, Duke Mathematical Journal 3, 443–454 (1937)

    work page 1937

  3. [3]

    Cam- bridge University Press, Cambridge (1998)

    Clark, D.M., Davey, B.A.: Natural Dualities for the Working Algebraist. Cam- bridge University Press, Cambridge (1998)

    work page 1998

  4. [4]

    Acta Uni- versitatis Matthiae Belii, series Mathematics 30, 1–35 (2022)

    Craig, A.P.K.: Representations and dualities for bounded lattices. Acta Uni- versitatis Matthiae Belii, series Mathematics 30, 1–35 (2022). Available at https://actamath.savbb.sk/pdf/aumb3001.pdf

    work page 2022

  5. [5]

    Algebra Universalis 74, 123–138 (2015)

    Craig, A.P.K., Gouveia M.J., Haviar, M.: TiRS graphs and TiRS frames: a new setting for duals of canonical extensions. Algebra Universalis 74, 123–138 (2015)

    work page 2015

  6. [6]

    Algebra Universalis 83:12 (2022)

    Craig, A.P.K., Gouveia M.J., Haviar, M.: Canonical extensions of lattices are more than perfect. Algebra Universalis 83:12 (2022)

    work page 2022

  7. [7]

    Craig, A., Haviar, M.: Reconciliation of approaches to the construction of canonical extensions of bounded lattices. Math. Slovaca 64, 1335–1356 (2014)

    work page 2014

  8. [8]

    Craig, A.P.K., Haviar, M., Priestley, H.A.: A fresh perspective on the canonical extensions of bounded lattices. Appl. Categ. Struct. 21, 725–749 (2013)

    work page 2013

  9. [9]

    CUBO, A Mathematical Journal 24, 369–392 (2022)

    Craig, A.P.K., Haviar, M., S˜ ao Jo˜ ao, J.: Dual digraphs of finite semidistributive lattices. CUBO, A Mathematical Journal 24, 369–392 (2022)

    work page 2022

  10. [10]

    CUBO, A Mathematical Journal 26, 279–302 (2024)

    Craig, A.P.K., Haviar, M., Marais, K.: Dual digraphs of finite meet-distributive and modular lattices. CUBO, A Mathematical Journal 26, 279–302 (2024)

    work page 2024

  11. [11]

    Algebra Universalis 5, 72–75 (1975)

    Davey, B.A., Poguntke, W., Rival, I.: A characterization of semi-distributivity. Algebra Universalis 5, 72–75 (1975)

    work page 1975

  12. [12]

    In: Contributions to Lattice Theory (Szeged, 1980), (A.P

    Davey, B.A., Werner, H.: Dualities and equivalences for varieties of algebras. In: Contributions to Lattice Theory (Szeged, 1980), (A.P. Huhn and E.T. Schmidt, eds). Coll. Math. Soc. J´ anos Bolyai33, North-Holland, Amsterdam, pp. 101– 275 (1983)

    work page 1980

  13. [13]

    2nd edition (2003)

    Gr¨ atzer, G.: General Lattice Theory. 2nd edition (2003)

    work page 2003

  14. [14]

    J´ onsson, B.: Sublattices of a free lattice. Canad. J. Math. 13, 256–264 (1961)

    work page 1961

  15. [15]

    and Rival, I.: Lattice varieties covering the smallest non-modular variety

    J´ onsson, B. and Rival, I.: Lattice varieties covering the smallest non-modular variety. Pacific J. Math. 82, 463–478 (1979)

    work page 1979

  16. [16]

    Tatra Mountains Math

    Ploˇ sˇ cica, M.: A natural representation of bounded lattices. Tatra Mountains Math. Publ. 5, 75–88 (1995)

    work page 1995

  17. [17]

    Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)

    work page 1970

  18. [18]

    Priestley, H.A.: Ordered topological spaces and the representation of distribu- tive lattices. Proc. London Math. Soc. 24, 507–530 (1972) Dual spaces of lattices and semidistributive lattices 27

    work page 1972

  19. [19]

    Algebra Univer- salis 8, 45–58 (1978)

    Urquhart, A.: A topological representation theory for lattices. Algebra Univer- salis 8, 45–58 (1978)

    work page 1978

  20. [20]

    Whitman, P.M.: Free lattices. Ann. Math. 42, 325–330 (1941) Andrew Craig Department of Mathematics and Applied Mathematics University of Johannesburg PO Box 524, Auckland Park, 2006 South Africa and National Institute of Theoretical and Computational Sciences (NITheCS) Johannesburg South Africa URL: https://orcid.org/0000-0002-4787-3760 e-mail: acraig@uj....

    work page 1941