Canonical extensions of lattices are more than perfect
Abstract
In CGH15 we introduced TiRS graphs and TiRS frames to create a new natural setting for duals of canonical extensions of lattices. In this continuation of CGH15 we answer Problem 2 from there by characterising the perfect lattices that are dual to TiRS frames (and hence TiRS graphs). We introduce a new subclass of perfect lattices called PTi lattices and show that the canonical extensions of lattices are PTi lattices, and so are `more' than just perfect lattices. We introduce morphisms of TiRS structures and put our correspondence between TiRS graphs and TiRS frames from CGH15 into a full categorical framework. We illustrate our correspondences between classes of perfects lattices and classes of TiRS graphs by examples.
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