Rational relations for modelling and analyzing LTI systems

If M is an R-module over an abelian ring R, then the set of all total submodules of M² is a seminearring (T, + , ·), where (+) is relation addition, and (·) is composition. If B is a Bezout domain of linear surjections on M, we construct a subseminearring Q of T consisting of so-called rational relations on M. An example is the set Q of single-input single-output relations defined by linear time-invariant (LTI) differential equations. A subseminearring of this Q is the field F of transfer functions, which approximate such relations as operators by neglecting their free response. Since rational relations include the free response, we propose using them instead of transfer functions to model and analyze LTI systems. Connections to results in behavioral systems theory are described.