Asian Journal of Pure and Applied Mathematics

The paper discusses the restriction of the theory of classical lattice isomorphism to the analysis of algebraic systems, in which current results are mostly restricted to modules over rings of polynomials and therefore to the analysis of causal systems. The main omission is that there is no single framework which can manage modules across rational function rings, and these are required to model noncausal, multidimensional, and distributed dynamical systems. In order to fill this gap, the paper constructs a lattice-theoretic generalised framework of finitely generated modules over the rings of rational functions in terms of module theory, localisation and bilinear forms. It is a combination of structural analysis of submodules, annihilator theory and duality by non-degenerate bilinear mappings to generalise classical results. The principal finding is that, given a non-degeneracy condition, there is a lattice isomorphism (up to duality) between the lattice of submodules of a module over a rational function ring and the submodule lattice of its annihilator. This theorem is a generalisation of the classical lattice isomorphism by Fuhrmann between polynomials and rationals, with all the necessary lattice operations maintained but allowing a more arbitrary algebraic structure. It has been extended to multidimensional dynamical systems, in which it allows the study of noncausal and spatially distributed phenomena. Control theory, coding theory and signal processing have practical implications where rational representations give a better model of the interconnections of complex systems and system structure.