Chapman University

Consider the following simple question:

Is there a subcategory of Top that is dually equivalent to Lat?

where Top is the category of topological spaces and continuous maps and Lat is the category

of bounded lattices and bounded lattice homomorphisms.

Of course, the question has been answered positively by specializing Lat, and (less

well-known) by generalizing Top.

The earliest examples are of the former sort: Tarski showed that every complete atomic

Boolean lattice is represented by a powerset (discrete topological space); Birkhoff showed

that every finite distributive lattice is represented by the lower sets of a finite partial order

(finite T0 space); Stone generalized Tarski and then Birkhoff, for arbitrary Boolean and

arbitrary bounded distributive lattices respectively. All of these results specialize Lat,

obtaining a (not necessarily full) subcategory of Top.

As a conceptual bridge, Priestley showed that distributive lattices can also be dually

represented in a category of certain topological spaces augmented with a partial order.

This is an example of the latter sort of result, namely, a duality between a category of

lattices and a subcategory of a generalization of Top.

Urquhart, Hartung and Hartonas developed dualities for arbitrary bounded lattices in

the spirit of Priestley duality, in that the duals are certain topological spaces equipped with

additional structure.

We take a different path via purely topological considerations. At the end, we obtain

an affirmative answer to the original question, plus a bit more, with no riders: the dual

categories to Lat and SLat (semilattices) are certain easily described subcategories of Top

simpliciter. This leads directly to a very natural topological characterization of canonical

extensions for arbitrary bounded lattices.

Building on the topological foundation, we consider lattices expanded with quasioperators,

i.e., operations that suitably generalize normal modal operatos, residuals, orthocomplements

and the like. This hinges on both the duality for lattices and for semilattices

in a natural way.

This talk is based on joint work with Peter Jipsen.

Date: May 2010.

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