Boolean and distributive ordered sets: Characterization and representation by sets
Let \((A,\leq)\) be an ordered set and \(X\subseteq A\). Denote by \(L(X)\) the lower cone of \(X\) in \(A\) and by \(U(X)\) the upper cone of \(X\). An ordered set \((A, \leq)\) is said to be distributive if \(L(x, U(y,z))= L(U(L (x,y),L(x,z)))\) for all \(x,y,z\in A\). An element \(y\in A\) is called a complement of \(x\) in \((A,\leq)\) if \(L(x,y)= L(A)\) and \(U(x,y)= U(A)\). A Boolean ordered set is defined as a distributive ordered set of which each element has a complement in \(A\). Some well-known theorems such as the Glivenko-Stone theorem and the Stone representation theorem on Boolean algebras and the Birkhoff representation theorem for distributive lattices are generalized to Boolean ordered sets and to distributive ordered sets, respectively.
scientific article; zbMATH DE number 828190