Double approximation and complete lattices (Q2882375)

The authors define double approximation systems as a generalization of approximation spaces and complete prime lattices as a generalization of complete atomic Boolean algebras. For a set \(X\) and two equivalence relations \(R\), \(S\) on \(X\), a triplet \((X,S,R)\) is called a double approximation system. A complete lattice \(L\) is called complete prime if a set \(\mathcal{J}_p(L)\) of completely join-prime elements is join-dense in \(L\) and a set \(\mathcal{M}_p(L)\) of completely meet-prime elements in \(L\) is meet-dense in \(L\). They prove a representation theorem of complete prime lattices by double approximation systems:NEWLINENEWLINETheorem 3.1 Let \(L\) be a complete prime lattice and \(G(L)=(X_L, \pi_0, \pi_1)\) be a double approximation system obtained by \(L\). A map \(\epsilon_{L,1} : L \to \operatorname{Fix}(\pi_{0 *} \pi_1^*)\) defined by \(\epsilon_{L,0} (x) = \pi_0^{-1} (\downarrow x)\) is an order isomorphism. Hence \(L\) is isomorphic to \(\operatorname{Fix}(\pi_{0 *} \pi_1^*)\) as a complete lattice.