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In mathematics, duality theory for distributive lattices provides three different representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone, generalizes the well-known Stone duality between Stone spaces and Boolean algebras.
Let L be a bounded distributive lattice, and let X denote the set of prime filters of L. For each a ∈ L, let φ+ = {x∈ X : a ∈ x}. Then is a spectral space, where the topology τ+ on X is generated by {φ+ : a ∈ L}. The spectral space is called the prime spectrum of L.
The map φ+ is a lattice isomorphism from L onto the lattice of all compact open subsets of. In fact, each spectral space is homeomorphic to the prime spectrum of some bounded distributive lattice.
Similarly, if φ− = {x∈ X : a ∉ x} and τ− denotes the topology generated by {φ− : a∈ L}, then is also a spectral space. Moreover, is a pairwise Stone space. The pairwise Stone space is called the bitopological dual of L. Each pairwise Stone space is bi-homeomorphic to the bitopological dual of some bounded distributive lattice.