Universal Algebra Seminar

Ross Willard, Department of Pure Mathematics, University of Waterloo

"Dualizing structures that are necessarily of infinite signature"

Many classical dualities, such as Stone duality for Boolean algebras and Priestley duality for distributive lattices, are mediated by a “schizophrenic pair” consisting of a finite algebra generating the algebraic category together with a topological structure having the same universe as, and compatible with, the finite generating algebra. For example, the schizophrenic pair for Stone duality consists of the 2-element Boolean algebra together with the 2-element set viewed as a discrete space, while the schizophrenic pair for Priestley duality consists of the 2-element distributive lattice together with the 2-element total order (again viewed as a discrete space). A long-standing open problem in the field of natural dualities asked if every schizophrenic pair is equivalent to one in which the topological structure has only finitely many operations and relations in its signature.

This problem was finally solved by Jane Pitkethly in 2011. In this talk I will describe the schizophrenic pairs that she constructed for this purpose and begin their analysis.

MC 5403