Joseph McDonald

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.

A class of ordered relational topological spaces is described, which we call orthomodular spaces. Our construction of these spaces involves adding a topology to the class of orthomodular frames introduced by Hartonas, along the lines of Bimbó's topologization of the class of orthoframes employed by Goldblatt in his representation of ortholattices. We then prove that the category of orthomodular lattices and homomorphisms is dually equivalent to the category of orthomodular spaces and certain continuous frame morphisms, which we call continuous weak p‐morphisms. It is well‐known that orthomodular lattices provide an algebraic semantics for the quantum logic. Hence, as an application of our duality, we develop a topological semantics for using orthomodular spaces and prove soundness and completeness.