Lattice isomorphisms between certain sublattices of continuous functions

Abstract

Let C(X, I) be the lattice of all continuous functions on a compact Hausdorff space X with values in the unit interval I = [0, 1]. We show that for compact Hausdorff spaces X and Y and (not necessarily contain constants) sublattices A and B of C(X, I) and C(Y, I), respectively, which satisfy a certain separation property, any lattice isomorphism ϕ : A −→ B induces a homeomorphism µ : Y −→ X. If, furthermore, A and B are closed under the multiplication, then ϕ has a representation ϕ(f )(y) = m y (f (µ(y))), f ∈ A, for all points y in a dense G δ subset Y 0 of Y , where each m y is a strictly increasing continuous bijection on I. In particular, for the case where X and Y are metric spaces and A and B are the lattices of all Lipschitz functions with values in I, the set Y 0 is the whole of Y .