Dual Plo\v{s}\v{c}ica spaces of ortholattices

TL;DR

This paper develops a dual representation theory for ortholattices using dual Ploica spaces, generalizing Priestley spaces and providing new duality theorems for ortholattices.

Contribution

It introduces the concept of dual Ploica spaces for ortholattices and establishes dual representation theorems, extending existing lattice dualities.

Findings

Improved definition of Ploica spaces for total order disconnectedness.

Defined dual space of an ortholattice via dual Ploica spaces.

Established dual representation theorems for ortholattices.

Abstract

We describe digraphs with topology which give dual representations of ortholattices. This is done via so-called dual Plo\v{s}\v{c}ica spaces of lattices. First, we improve the definition of Plo\v{s}\v{c}ica 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 Plo\v{s}\v{c}ica space of the lattice-reduct of the ortholattice equipped with a map representing the orthocomplement operation. We introduce an abstract ortho-Plo\v{s}\v{c}ica space capturing the properties of the dual space of an ortholattice, and we present dual representation theorems between general ortholattices and the ortho-Plo\v{s}\v{c}ica spaces. We illustrate our dual representations by examples.