Abstract. Let X and Y be compact Hausdorff spaces, E be a Banach lattice and F be an AM space with unit. Let π : C(X, E) → C(Y, F ) be a Riesz isomorphism such that 0 ∈ f (X) if and only if 0 ∈ π(f )(Y ) for each f ∈ C(X, E). We prove that X is homeomorphic to Y and E is Riesz isomorphic to F . This generalizes some known results.

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Let E and F be uniformly complete vector lattices with disjoint complete systems (ui)i∈I and (vi)i∈I of projection elements of E and F respectively. In this paper we prove that if T is a lattice homomorphism from E into F with T (λui) = λvi for each λ ∈ R and i ∈ I then T is linear. This generalizes the main results of [4] and [5].We refer to the standard texts [1] and [2] for the definitions and notations. Throughout this paper vector lattices are assumed to be Archimedean.A map T between vector lattices E and F is called a lattice homomorphism ifA linear lattice homomorphism is called a Riesz homomorphism. As usual C(X) denotes the vector lattice of real valued continuous functions defined on a topological space X under point-wise operations.In [4] Mena and Roth proved the following: Let K and M be compact Hausdorff spaces and let T : C(K) → C(M ) be a lattice homomorphism with T (λ1) = λ1 for each λ ∈ R. Then T is linear. In [5], it was shown that the compact Hausdorff space can be replaced by a real compact Hausdorff space and also the theorem was generalized in [3] as T is linear if and only if T is linear on constant functions.Using the Kakutani representation theorem the Mena and Roth theorem can be stated as follows: Let E 1 and E 2 be uniformly complete vector lattices with order units e 1 and e 2 , respectively, and let T : E 1 → E 2 be a lattice homomorphism with T (λe 1 ) = λe 2 for each λ ∈ R. Then T is linear. Now we can generalize the Theorem of Mena and Roth as follows:Lemma 1 Let E 1 and E 2 be uniformly complete vector lattices with weak order units e 1 and e 2 , respectively. If T is a lattice homomorphism from E 1 into E 2 satisfying T (λe 1 ) = λe 2 for each λ ∈ R then T is linear.

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We introduce the notion of unbounded locally solid Riesz spaces, and investigate its fundamental properties.