## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

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 x43484

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

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.