Ordered separation axioms and the Wallman ordered compactification

Two constructions have been given previously of the Wallman ordered compactification w₀X of a T₁-ordered, convex ordered topological space (X, τ, ≤). Both of those papers note that w ₀X is T₁, but need not be T₁-ordered. Using this as one motivation, we propose a new version of T₁-ordered, called T₁^K-ordered, which has the property that the Wallman ordered compactification of a T₁^K-ordered topological space is T₁^K-ordered. We also discuss the R₀-ordered (R₀^K-ordered) property, defined so that an ordered topological space is T₁-ordered (T₁ ^K-ordered) if and only if it is T₀-ordered and R ₀-ordered (R₀^K-ordered).