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Dual to bounded distributive lattices are Priestley spaces, and that for Heyting algebras are Esakia spaces. Do we have such results for the class of lattices which are not bounded in general, preferably for distributive lattices or contrapositionally complemented algebras (the class of algebra corresponding to minimal logic)?

My understanding: The set of clopen sets of a Priestley space forms a bounded distributive lattice. Now, empty set and $X$ are always the smallest and largest clopen sets respectively in any topological space over $X$. Thus the lattice of the set of clopen sets will always be bounded. So, just taking the set of clopen sets will not work for unbounded case.

Anuj More
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1 Answers1

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The answer is yes: the category of distributive lattices is dually equivalent to the category of bounded Priestley spaces.

A bounded Priestley space is a tuple $(X,0,1,\leq,\tau)$ where $(X,\leq,\tau)$ is a Priestley space and the order $\leq$ on $X$ has a bottom element $0$ and top element $1$.

In fact$^1$, every distributive lattice is isomorphic to the lattice of all proper, nonempty, clopen, decreasing subsets of some bounded Priestley space.

If you have a bounded Priestley space with no atoms (i.e. elements covering $0$), then the nonempty, clopen, decreasing subsets do not necessarily have a least element.

$^1$This is proven in Clark and Davey's Natural Dualities for the Working Algebraist.

Eran
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