Priestley duality for order-preserving maps into distributive lattices (Q1815850)

Let \(R\) be a binary relation between Priestley spaces \(P\) and \(Q\). Then \(R\) is called a Priestley relation if for all \(p\in P\), \(R(p)= \{q\mid (p,q)\in R\}\) is a closed down-set; and for all \(V\in D(Q)\), \(R^{-1}(V)\in D(P)\). Here \(D(P)\) denotes the lattice of clopen up-sets. The author shows that the category of bounded distributive lattices with order-preserving maps is dually equivalent to the category of Priestley spaces with Priestley multirelations. The author also investigates the Priestley dual of the lattice of all continuous order-preserving maps from a poset \(P\) into a bounded distributive lattice \(L\), where \(L\) carries the discrete topology.