Enumeration of classes of linear systems via equations and via partitions in an ordered abelian monoid

We prove that the feedback classes of locally Brunovsky linear systems are given by the decomposition of state space into direct summands. Hence the problem of enumerating feedback classes of systems is equivalent to the problem of enumerating all solutions of a linear equation over a commutative zerosumfree monoid. If this monoid is also cancelative then the problem is solved by all the partitions.