Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dual Ploščica spaces of ortholattices

Andrew Craig, Miroslav Haviar

TL;DR

The paper develops a refined duality framework for ortholattices by introducing dual Ploščica spaces, which generalize Priestley spaces through the TED (total edge-disconnected) condition. It then extends to ortho-Ploščica spaces by incorporating a continuous $g$-map that encodes orthocomplementation, enabling robust dual representation theorems between general ortholattices and these spaces. The authors prove that every ortholattice is dual to an appropriate ortho-Ploščica space and that the dual of such a space recovers the original lattice, with the $g$-operation preserved under duality. Concrete examples, including the 96-element free orthomodular lattice and two infinite lattices, illustrate the construction and topology of the duals, highlighting the applicability of the approach to quantum-logical structures.

Abstract

We describe digraphs with topology which give dual representations of ortholattices. This is done via so-called dual Ploščica spaces of lattices. First, we improve the definition of Ploščica spaces from an earlier paper to give a straight and natural generalisation of the total order disconnectedness of Priestley spaces. Then we define the dual space of a general ortholattice as the dual Ploščica space of the lattice-reduct of the ortholattice equipped with a map representing the orthocomplement operation. We introduce an abstract ortho-Ploščica space capturing the properties of the dual space of an ortholattice, and we present dual representation theorems between general ortholattices and the ortho-Ploščica spaces. We illustrate our dual representations by examples.

Dual Ploščica spaces of ortholattices

TL;DR

The paper develops a refined duality framework for ortholattices by introducing dual Ploščica spaces, which generalize Priestley spaces through the TED (total edge-disconnected) condition. It then extends to ortho-Ploščica spaces by incorporating a continuous -map that encodes orthocomplementation, enabling robust dual representation theorems between general ortholattices and these spaces. The authors prove that every ortholattice is dual to an appropriate ortho-Ploščica space and that the dual of such a space recovers the original lattice, with the -operation preserved under duality. Concrete examples, including the 96-element free orthomodular lattice and two infinite lattices, illustrate the construction and topology of the duals, highlighting the applicability of the approach to quantum-logical structures.

Abstract

We describe digraphs with topology which give dual representations of ortholattices. This is done via so-called dual Ploščica spaces of lattices. First, we improve the definition of Ploščica spaces from an earlier paper to give a straight and natural generalisation of the total order disconnectedness of Priestley spaces. Then we define the dual space of a general ortholattice as the dual Ploščica space of the lattice-reduct of the ortholattice equipped with a map representing the orthocomplement operation. We introduce an abstract ortho-Ploščica space capturing the properties of the dual space of an ortholattice, and we present dual representation theorems between general ortholattices and the ortho-Ploščica spaces. We illustrate our dual representations by examples.
Paper Structure (6 sections, 19 theorems, 17 equations, 4 figures, 3 tables)

This paper contains 6 sections, 19 theorems, 17 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplifying Ploščica spaces
  4. Ortho-Ploščica spaces
  5. Dual representation theorems
  6. Examples

Key Result

Proposition 2.3

Let $G=(V,E)$ be a TiRS digraph. Then

Figures (4)

  • Figure 1: The dual digraph of $\mathcal{F}_{\mathcal{OM}}(2)$. The $g$-map is indicated with the dotted arrows.
  • Figure 2: The infinite ortholattice $\mathbf{O}_{\mathbb{Z}}$ and its dual digraph. The involutive map $g$ is indicated with dotted arrows. Not all edges of the digraph are included. Refer to Table \ref{['table:digraph-of-OZ']} for a complete description.
  • Figure 3: The infinite orthomodular lattice $\mathbf{M}_{\infty}$.
  • Figure 4: A part of the dual digraph of $\mathbf{M}_\infty$ (six vertices with distinct $j,k,l \in \mathbb{Z}{\setminus}\{0\}$).

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2: P3
  • Proposition 2.3: P6
  • Lemma 2.4: cf. Plos95
  • Proposition 2.5: Plos95, P6
  • Lemma 2.6
  • proof
  • Lemma 2.7: P6
  • Definition 3.1: P6
  • Proposition 3.2: P6
  • ...and 29 more