PhD student Joseph McDonald is awarded SSHRC CGS-D and President's Doctoral Prize of Distinction
16 December 2022
PhD student Joseph McDonald has been awarded the SSHRC CGS-D and the President's Doctoral Prize of Distinction for his work on the duality theory of ordered algebraic structures and its applications. Here is a description of his research:
My research investigates an interaction that occurs between algebra, topology, and logic. This is an area known as “Stone duality” and is named in honour of M. H. Stone who in [3] established the foundations for the now well-known result that the category of Boolean algebras and Boolean homomorphisms is dually equivalent to the category of Stone spaces and continuous functions. Nowadays, there is an ample supply of dualities linking categories of partially ordered algebras and categories of topological spaces.
Throughout the tenure of my CGS-D award, I shall first investigate the duality theory of orthomodular and modular ortholattices under the duality that was recently established for general ortholattices by myself and K. Yamamoto in [2]. These are the two most important subclasses of ortholattices and arise as the Lindenbaum algebras of various quantum logics. After the desired results are obtained, I shall then study the duality theory of these algebras equipped with additional operators, such as Sasaki projection, Sasaki coprojection, as well as certain modal operators, using the algebraic methods of gaggle theory developed by Bimbó and Dunn in [1].
The first application of the duality results I hope to obtain will be aimed at providing topological characterizations of important constructions in the algebraic theory of these lattices. Second, by exploiting the well-known algebraic semantics these lattices provide for various quantum logics, I also aim to prove topological soundness and completeness theorems for such quantum logics and investigate their model-theoretic properties. This work will be conducted under the research supervision of Dr. Katalin Bimbó.
[1] Bimbó, K., Dunn J. M.: Generalized Galois Logics: Relational Semantics of Non-Classical Logical Calculi. CSLI Lecture Notes vol. 188. CSLI Publications, Stanford, 2008
[2] McDonald, J., Yamamoto, K.: Choice-free duality for orthocomplemented lattices by means of spectral space. Algebra Universalis, 83(4), 2022
[3] Stone, M. H.: The theory of representations of Boolean algebras. Transactions of the American Mathematical Society, 40(1), 1936