Ross Willard, Pure Mathematics, University of Waterloo
"Natural dualities for finitely generated quasi-varieties - definitions and first results"
I will present the basic definitions of natural duality theory developed by David Clark and Brian Davey. One begins with a quasi-variety ISP($\mathbf{M}$) generated by a finite algebra $\mathbf{M}$, and chooses a relational structure $\mathbb{M}$ with the same universe as $\mathbf{M}$, endowed with the discrete topology. For any algebra $\mathbf{A}$ belonging to the quasi-variety ISP($\mathbf{M}$), one can naturally interpret $\mathbb{X} := \mathrm{Hom}(\mathbf{A},\mathbf{M}$) as a structured Stone space living in a common category with $\mathbb{M}$. $\mathbb{X}$ is the \emph{dual} of $\mathbf{A}$. What is desired is that $\mathrm{Hom}(\mathbb{X},\mathbb{M})$, the \emph{double dual} of $\mathbf{A}$, be naturally interpretable as an algebra isomorphic to $\mathbf{A}$ via the natural evaluation map. Whether or not this is true (for all $\mathbf{A}$) depends on our choice of $\mathbb{M}$ (and, as we will see in future lectures, is impossible for some $\mathbf{M}$). In this lecture I will define these notions precisely and state some basic properties.