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Duality Theory for Bounded Lattices: A Comparative Study

Guram Bezhanishvili, Luca Carai, Patrick Morandi

TL;DR

The paper provides a comprehensive, unified comparison of non-distributive dualities for bounded lattices by recasting morphisms as relations and building a network of category equivalences among major approaches (JM, BDGM, CG, Hartonas, DH, GvG, Hg, Urq, Plo). It shows how BDGM-spaces, JM-spaces, and related frameworks relate to the Lat/ SLat dualities via precise functors and relational morphisms, clarifying how these theories interlock and how the distributive case collapses to Priestley duality. A central contribution is the demonstration that a circle of equivalences DH ≃ GvG ≃ Hg ≃ Urq ≃ Plo ≃ Lat can be established with explicit natural isomorphisms, thereby unifying diverse dualities under a single relational paradigm. The results also illuminate how Dunn-Hartonas can recover Hartonas and BDGM dualities and how the older Urquhart-Hartung-Ploščica approaches fit into the broader framework, ensuring a robust, interconnected toolkit for non-distributive lattice duality. In short, the work provides a cohesive algebraic-topological atlas for bounded lattices beyond distributivity, enabling translation of results across multiple dualities and clarifying the precise morphisms that correspond to lattice homomorphisms.

Abstract

There are numerous generalizations of the celebrated Priestley duality for bounded distributive lattices to the non-distributive setting. The resulting dualities rely on an earlier foundational work of such authors as Nachbin, Birkhoff-Frink, Bruns-Lakser, Hofmann-Mislove-Stralka, and others. We undertake a detailed comparative study of the existing dualities for arbitrary bounded (non-distributive) lattices, including supplying the dual description of bounded lattice homomorphisms where it was lacking. This is achieved by working with relations instead of functions. As a result, we arrive at a landscape of categories that provide various generalizations of the category of Priestley spaces. We provide explicit descriptions of the functors yielding equivalences of these categories, together with explicit equivalences with the category of bounded lattices and bounded lattice homomorphisms.

Duality Theory for Bounded Lattices: A Comparative Study

TL;DR

The paper provides a comprehensive, unified comparison of non-distributive dualities for bounded lattices by recasting morphisms as relations and building a network of category equivalences among major approaches (JM, BDGM, CG, Hartonas, DH, GvG, Hg, Urq, Plo). It shows how BDGM-spaces, JM-spaces, and related frameworks relate to the Lat/ SLat dualities via precise functors and relational morphisms, clarifying how these theories interlock and how the distributive case collapses to Priestley duality. A central contribution is the demonstration that a circle of equivalences DH ≃ GvG ≃ Hg ≃ Urq ≃ Plo ≃ Lat can be established with explicit natural isomorphisms, thereby unifying diverse dualities under a single relational paradigm. The results also illuminate how Dunn-Hartonas can recover Hartonas and BDGM dualities and how the older Urquhart-Hartung-Ploščica approaches fit into the broader framework, ensuring a robust, interconnected toolkit for non-distributive lattice duality. In short, the work provides a cohesive algebraic-topological atlas for bounded lattices beyond distributivity, enabling translation of results across multiple dualities and clarifying the precise morphisms that correspond to lattice homomorphisms.

Abstract

There are numerous generalizations of the celebrated Priestley duality for bounded distributive lattices to the non-distributive setting. The resulting dualities rely on an earlier foundational work of such authors as Nachbin, Birkhoff-Frink, Bruns-Lakser, Hofmann-Mislove-Stralka, and others. We undertake a detailed comparative study of the existing dualities for arbitrary bounded (non-distributive) lattices, including supplying the dual description of bounded lattice homomorphisms where it was lacking. This is achieved by working with relations instead of functions. As a result, we arrive at a landscape of categories that provide various generalizations of the category of Priestley spaces. We provide explicit descriptions of the functors yielding equivalences of these categories, together with explicit equivalences with the category of bounded lattices and bounded lattice homomorphisms.

Paper Structure

This paper contains 22 sections, 129 theorems, 170 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sketch of proof.
  4. The Jipsen-Moshier approach
  5. The Bezhanishvili-Dmitrieva-de Groot-Moraschini approach
  6. The Hartonas approach
  7. The Celani-González approach
  8. The Dunn-Hartonas approach
  9. From Dunn-Hartonas to Gehrke-van Gool
  10. From Gehrke-van Gool to Hartung
  11. From Hartung to Urquhart
  12. Ploščica spaces: an alternative approach to Urquhart spaces
  13. From Urquhart back to Dunn-Hartonas
  14. The resulting category equivalences
  15. The natural isomorphism Lg
  16. ...and 7 more sections

Key Result

Theorem 2.1

${\sf{Spec}}$ is isomorphic to ${\sf{Pries}}$.

Figures (2)

  • Figure 1: Diagram of the equivalences
  • Figure 2: Various dualities for ${\sf Lat}$

Theorems & Definitions (305)

  • Theorem 2.1: Cornish's isomorphism Cor75
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5: Nachbin's theorem Nac49
  • Definition 2.7
  • Theorem 2.8
  • Definition 2.9
  • Theorem 2.10
  • Remark 2.11
  • ...and 295 more