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  "body": "\n## Statement\n\nLet $L$ be a bounded [[distributive lattice|0HM0HQCK6N]].\nA *prime filter* of $L$ is a [[filter|Z15FXM9WM1]] $F$ with $0 \\notin F$ such that $a \\vee b \\in F$ implies $a \\in F$ or $b \\in F$.\nLet $X_L$ be the set of prime filters of $L$, ordered by inclusion.\nGive $X_L$ the topology generated by the sets $\\set{F : a \\in F}$ and $\\set{F : a \\notin F}$ for $a \\in L$.\nFor a [[Priestley space|ZGBV8V3823]] $X$, let $\\mathrm{ClopUp}(X)$ be the lattice of subsets of $X$ that are clopen [[upper sets|ETCE287E2R]].\nIts meet and join are intersection and union.\n\n1. $X_L$ is a Priestley space, and the map $a \\mapsto \\set{F \\in X_L : a \\in F}$ is an isomorphism of bounded lattices from $L$ onto $\\mathrm{ClopUp}(X_L)$.\n2. For every Priestley space $X$, the map $x \\mapsto \\set{U \\in \\mathrm{ClopUp}(X) : x \\in U}$ is a homeomorphism and order isomorphism from $X$ onto $X_{\\mathrm{ClopUp}(X)}$.\n3. These constructions extend to a dual equivalence of categories.\nOn one side are the bounded distributive lattices with bounded lattice homomorphisms; on the other, the Priestley spaces with continuous order-preserving maps.\nA homomorphism $h \\colon L \\to M$ corresponds to the map $X_M \\to X_L$, $G \\mapsto h^{-1}(G)$.\n\n## Remarks\n\nThe key input is the [[prime ideal theorem for distributive lattices|ME8GM0VTWP]]: if $a \\not\\le b$, some prime filter contains $a$ and not $b$.\nThis gives the injectivity in part 1 and the separation axiom of $X_L$.\nCompactness of $X_L$ comes from viewing $X_L$ as a closed subspace of the product space $\\set{0, 1}^L$.\n\nExample: if $L$ is finite, the prime filters are the sets ${\\uparrow} j$ for $j$ join-irreducible, and $X_L$ is a finite poset with the discrete topology.\nThen part 1 becomes [[Birkhoff's representation theorem|6RDM3069KC]] up to reversing the order.\n\nWhen $L$ is a [[Boolean algebra|0XADJKM699]], every prime filter is an ultrafilter and the order on $X_L$ is trivial.\nThen the duality restricts to Stone duality between Boolean algebras and [[Stone spaces|ZV38HA5PAQ]].\n\n## Explanations\n\n### L1 human\n\nEvery bounded distributive lattice can be drawn as a lattice of sets: the clopen upper sets of a special ordered space.\nConversely, from such a space one gets back the lattice, and from the lattice the space.\nSo questions about distributive lattices can be translated into questions about ordered compact spaces, and back.\n\n### L2 human\n\nPriestley duality is a widely used form of duality for distributive lattices.\nHomomorphisms turn into continuous order-preserving maps in the opposite direction, so algebraic questions become geometric ones.\nIt extends Stone duality for Boolean algebras.\nUnder it, Heyting algebras correspond to [[Esakia spaces|996BWM6ZQ7]], a special class of Priestley spaces.\nAn equivalent version, due to Stone, uses [[spectral spaces|G3W2WXY2VN]] instead of Priestley spaces.\n",
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