Canonical completion and duality for cylindric ortholattices and cylindric Boolean algebras

TL;DR

This paper explores the algebraic and topological duality theories for cylindric ortholattices and Boolean algebras, establishing canonical completions and spectral dualities that are constructively obtained without the Axiom of Choice.

Contribution

It demonstrates that cylindric ortholattices are closed under canonical completions and establishes a duality with spectral spaces, extending previous results to a constructive setting.

Findings

Cylindric ortholattices are closed under canonical completions.

A spectral topology duality is established for cylindric ortholattices.

Completion and duality results for cylindric Boolean algebras are obtained constructively.

Abstract

In this note, we investigate the algebraic and topological representation theory of cylindric ortholattices and cylindric Boolean algebras. The first contribution demonstrates that cylindric ortholattices are closed under canonical completions. By equipping a spectral topology to the dual space associated with the canonical completion, we then establish a dual equivalence between the category of cylindric ortholattices and a certain subcategory of the category of spectral spaces. This work builds on the completion and duality results obtained by Harding, McDonald, and Peinado in the setting of monadic ortholattices combined with the duality results obtained by McDonald and Yamamoto in the setting of general ortholattices. By working with the duality theory for Boolean algebras established by Bezhanishvili and Holliday, we then obtain completion and duality results for cylindric Boolean algebras. A key aspect of our duality results is that they are constructive in the sense that they obtain in Zermelo-Fraenkel set theory independently of the Axiom of Choice.