Topological duality and lattice expansions: canonial extensions via Stone duality
|
Wed, 26/05/2010 14:00 |
Drew Moshier (Chapman University) |
Analytic Topology in Mathematics and Computer Science  |
L3 |
| 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. 1 | |||