Characterizations of finite dimensional Archimedean vector lattices

In this paper, we give some necessary and sufficient conditions for an Archimedean vector lattice A to be of finite dimension. In this context, we give three characterizations. The first one contains the relation between the vector lattice A to be of finite dimension and its universal completion Au. The second one shows that the vector lattice A is of finite dimension if and only if one of the following two equivalent conditions holds : (a) every maximal modular algebra ideal in Au is relatively uniformly complete or (b) Orth(A,Au)=Z(A,Au) where Orth(A,Au) and Z(A,Au) denote the vector lattice of all orthomorphisms from A to Au and the sublattice consisting of orthomorphisms π with |π(x)|≤λ|x| (x∈A) for some 0≤λ∈R, respectively. It is well-known that any universally complete vector lattice A is of the form C∞(X) for some Hausdorff extremally disconnected compact topological space X. The point x∈X is called σ- isolated if the intersection of every sequence of neighborhoods of x is a neighborhood of x. The last characterization of finite dimensional Archimedean vector lattices is the following. Let A be a vector lattice and let Au(=C∞(X)) be its universal completion. Then A is of finite dimension if and only if each element of X is σ-isolated. Bresar in \cite{4} raised a question to find new examples of zero product determined algebras. Finally, as an application, we give a positive answer to this question.