Lattices of continuous monotonic functions

Let Y 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 〈X.I〉 has the Tietze extension property for order preserving continuous functions from X to I. Denote by M(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.