Research

My research interests are in universal algebra and ordered algebraic structures. In particular, I study commutative BCK-algebras, which are loosely “in between” Boolean algebras and distributive lattices, but have ties with many other structures: MV-algebras, lattice-ordered abelian groups, BCI-algebras, AF C*-algebras, Łukasiewicz algebras, commutative integral residuated lattices, and more. There is a notion of “prime ideal” for these algebras, and the set of prime ideals can be endowed the Zariski topology. This topological space is called the spectrum of the corresponding algebra. My work deals with the ideal theory and the spectral theory of these commutative BCK-algebras, as well as the interplay between the categories of commutative BCK-algebras, generalized spectral spaces, and distributive lattices (with 0). Some of this work generalizes to the class of all BCK-algebras, where we drop the assumption of commutativity.

Much of my work to this point can be (broadly) seen as attempting to answer two questions:

1. What topological spaces can occur as the prime spectrum of a commutative BCK-algebra?

2. What distributive lattices can occur as the lattice of ideals of a commutative BCK-algebra?

 Recently I have also been thinking about free cBCK-algebras, coproducts of cBCK-algebras, coproducts of generalized spectral spaces, varietal properties of quasicommutative BCK-algebras, and satisfiability degrees in finite BCK-algebras.

Papers on the arXiv: 

• Spectral properties of cBCK-algebras (Algebra Universalis, 83 (3), article 25, 2022)

• Satisfiability degree in BCK-algebras (Scientiae Mathematicae Japonicae, 36, 2023)

• Every complete atomic Boolean algebra is the ideal lattice of a cBCK-algebra  (European Journal of Mathematics, 10, article 40, 2024)

Here is my full research statement. (updated 10/30/2022)

Slides from a talk can be viewed here: Satisfiability degree in BCK-algebras (2022)

A copy of my dissertation can be found here, and some small errata can be found here