The Lattice of Ordered Compactifications of a Direct Sum of Totally Ordered Spaces

The lattice of ordered compactifications of a topological sum of a finite number of totally ordered spaces is investigated. This investigation proceeds by decomposing the lattice into equivalence classes determined by the identification of essential pairs of singularities. This lattice of equivalence classes is isomorphic to a power set lattice. Each of these equivalence classes is further decomposed into equivalence classes determined by admissible partially ordered partitions of the ordered Stone–Čech remainder. The lattice structure within each equivalence class is determined using an algorithm based on the incidence matrix of the partially ordered partition. As examples, the ordered compactification lattices for the spaces [0, 1) ⊕ [0, 1), [0, 1) ⊕ [0, 1) ⊕ [0, 1),R ⊕ R, and R\{0} ⊕R\{0} are determined.