Past Analytic Topology in Mathematics and Computer Science

26 May 2010
Drew Moshier
<p>Consider the following simple question:</p> <p>Is there a subcategory of Top that is dually equivalent to Lat?</p> <p>where Top is the category of topological spaces and continuous maps and Lat is the category</p> <p>of bounded lattices and bounded lattice homomorphisms.</p> <p>Of course, the question has been answered positively by specializing Lat, and (less</p> <p>well-known) by generalizing Top.</p> <p>The earliest examples are of the former sort: Tarski showed that every complete atomic</p> <p>Boolean lattice is represented by a powerset (discrete topological space); Birkhoff showed</p> <p>that every finite distributive lattice is represented by the lower sets of a finite partial order</p> <p>(finite T0 space); Stone generalized Tarski and then Birkhoff, for arbitrary Boolean and</p> <p>arbitrary bounded distributive lattices respectively. All of these results specialize Lat,</p> <p>obtaining a (not necessarily full) subcategory of Top.</p> <p>As a conceptual bridge, Priestley showed that distributive lattices can also be dually</p> <p>represented in a category of certain topological spaces augmented with a partial order.</p> <p>This is an example of the latter sort of result, namely, a duality between a category of</p> <p>lattices and a subcategory of a generalization of Top.</p> <p>Urquhart, Hartung and Hartonas developed dualities for arbitrary bounded lattices in</p> <p>the spirit of Priestley duality, in that the duals are certain topological spaces equipped with</p> <p>additional structure.</p> <p>We take a different path via purely topological considerations. At the end, we obtain</p> <p>an affirmative answer to the original question, plus a bit more, with no riders: the dual</p> <p>categories to Lat and SLat (semilattices) are certain easily described subcategories of Top</p> <p>simpliciter. This leads directly to a very natural topological characterization of canonical</p> <p>extensions for arbitrary bounded lattices.</p> <p>Building on the topological foundation, we consider lattices expanded with quasioperators,</p> <p>i.e., operations that suitably generalize normal modal operatos, residuals, orthocomplements</p> <p>and the like. This hinges on both the duality for lattices and for semilattices</p> <p>in a natural way.</p> <p>This talk is based on joint work with Peter Jipsen.</p> <p>Date: May 2010.</p> <p>1</p>
  • Analytic Topology in Mathematics and Computer Science