, Volume 31, Issue 3, pp 435–461 | Cite as

Distributive Envelopes and Topological Duality for Lattices via Canonical Extensions

  • Mai Gehrke
  • Samuel J. van Gool


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.


Lattice Non-distributivity Distributive envelope Canonical extension Priestley duality Pervin spaces Bicompletions 


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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.LIAFACNRS and Université Paris DiderotParis Cedex 13France
  2. 2.IMAPPRadboud Universiteit Nijmegen and LIAFA, Université Paris DiderotNijmegenThe Netherlands

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