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Register a new userDistributive envelopes and topological duality for lattices via canonical extensions
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Authors
Mai Gehrke
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We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them. This is a modification of a construction of the injective hull of a semilattice by Bruns and Lakser, adjusting their concept of admissibility to the finitary case. Finally, we show that the dual spaces of the distributive envelopes of a lattice coincide with completions of quasi-uniform spaces naturally associated with the lattice, thus giving a precise spatial meaning to the distributive envelopes.
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In cite{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 cite{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 cite{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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