Choice-free duality for orthocomplemented lattices by means of spectral spaces

TL;DR

This paper develops a choice-free topological representation and duality for orthocomplemented lattices using spectral spaces, avoiding nonconstructive principles like Alexander's Subbase Theorem and the Ultrafilter Theorem.

Contribution

It introduces a choice-free spectral space representation for orthocomplemented lattices and establishes a dual equivalence of categories without relying on the Axiom of Choice.

Findings

Provides a choice-free duality between orthocomplemented lattices and spectral spaces.

Defines upper Vietoris orthospaces to characterize spectral spaces of proper lattice filters.

Extends spectral space duality to orthocomplemented lattices without nonconstructive assumptions.

Abstract

The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets of a Stone space depends upon Alexander's Subbase Theorem, which asserts that a topological space $X$ is compact if every subbasic open cover of $X$ admits of a finite subcover. This is an easy consequence of the Ultrafilter Theorem - whose proof depends upon Zorn's Lemma, which is well known to be equivalent to the Axiom of Choice. Within this work, we give a choice-free topological representation of orthocomplemented lattices by means of a special subclass of spectral spaces; choice-free in the sense that our representation avoids use of Alexander's Subbase Theorem, along with its associated nonconstructive choice principles. We then introduce a new subclass of spectral spaces which we call \emph{upper Vietoris orthospaces} in order to characterize (up to homeomorphism and isomorphism) the spectral space of proper lattice filters used in our representation. It is then shown how our constructions give rise to a choice-free dual equivalence of categories between the category of orthocomplemented lattices and the dual category of upper Vietoris orthospaces. Our duality combines Bezhanishvili and Holliday's choice-free spectral space approach to Stone duality for Boolean algebras with Goldblatt and Bimb\'o's choice-dependent orthospace approach to Stone duality for orthocomplemented lattices.