Algebraic methods have penetrated deeply into contemporary complex analysis, having an essential influence on both the choice of problems and on the methods for solving them. This monograph deals with the applications of distributive lattices of subspaces to problems in multidimensional complex analysis. The author extends and refines Bogolyubov's 'edge-of-the-wedge' theorem, which is formulated as the exactness of a definite homology sequence in various classes of holomorphic functions. He also undertakes a systematic study of two classes of analytic functionals, the so-called Fourier hyperfunctions and the Fourier ultrahyperfunctions, on the basis of distributive lattices of linear subspaces. With this goal in mind, he obtains a homology characterization of distributive lattices of submodules, along with duality formulas for dual lattices of locally convex subspaces. The book is intended for specialists in the theory of functions of several complex variables.