Banach–Stone theorems and Riesz algebras

Let X, Y be compact Hausdorff spaces and let E, F be both Banach lattices and Riesz algebras. In this paper, the following main result shall be proved: If F has no zero-divisor and there exists a Riesz algebraic isomorphism such that Φ(f) has no zero if f has none, then X is homeomorphic to Y and E is Riesz algebraically isomorphic to F.