PhD student Joseph McDonald publishes research article in algebra journal

PhD student Joseph McDonald has published a research article in Algebra Universalis - an international mathematics journal focused on universal algebra, lattice theory, and related areas. The paper is titled “Choice-free duality for orthocomplemented lattices by means of spectral spaces” and was co-authored with K. Yamamoto of the Czech Academy of Sciences. Here is a description of their paper:

In this work, we study a certain class of algebras known as orthocomplemented lattices (ortholattices), which are generalizations of Boolean algebras. We prove a topological representation theorem which shows that every ortholattice is isomorphic to a certain algebra of subsets of a spectral space. We then introduce what we call upper Vietoris orthospaces and show that they characterize (up to relational homeomorphism) the spectrum of lattice filters used in our representation. The main theorem of our paper demonstrates that the category of ortholattices and their associated homomorphisms is dually equivalent to the category of upper Vietoris orthospaces and their associated continuous frame morphisms. Applications of our duality results are then explored within the algebraic theory of ortholattices. Unlike most Stone-type dualities for a given class of algebras, our duality obtains in the universe of Zermelo-Fraenkel set theory independently of the Axiom of Choice.