## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Wednesday, October 3, 2018 — 3:30 PM EDT

**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.

Location

MC 5403

,

Canada

,

Canada

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1