Lattices of continuous monotonic functions

Abstract:

Let $X$ be a compact Hausdorff space equipped with a closed partial ordering. Let $I$ be a linear ordering that either does not have a maximal element or does not have a minimal element. We further assume that $\langle {X,I} \rangle$ has the Tietze extension property for order preserving continuous functions from $X$ to $I$. Denote by $(X,I)$ the lattice of order preserving continuous functions from $X$ to $I$. We generalize a theorem of Kaplanski [K], and show that as a lattice alone, $M(X,I)$ characterizes $X$ as an ordered space.