Submodular Lattice Isomorphisms Between Some Modules Over the Ring of Rational Functions

Abstract

Fuhrmann’s work introduced a lattice isomorphism between polynomial submodules and closed formal series submodules, which plays a crucial role in the study of discrete linear systems within the behavioral framework. However, existing studies primarily focus on causal discrete systems, leaving a gap in the analysis of anti-causal systems. This paper extends Willems’s behavioral approach by establishing a lattice isomorphism between finitely generated submodules of the polynomial module and full-rank submodules of a free module over the ring of proper rational functions. The results provide a unifying algebraic structure that accommodates both causal and anti-causal systems. This generalization enhances the applicability of the behavioral framework and contributes to the ongoing development of algebraic system theory.