Nonvertical Affine Manifolds and Separation Theorems in Product Spaces

In the paper we introduce a general separation concept for sets in product spaces X × Z, where Z is partially ordered. Using some results about nonvertical affine manifolds we formulate some concerte separation theorems. Especially, we get assertions about the separation of sets by affine mappings, in which the order relation “≦” is replaced by the relation ??”?. Thus, these assertions are suitable to characterize PARETO optimal solutions of vector optimization problems.