Recurrence and finiteness for systems in a general setting

An X-system is G : I → Q, Fx : Q → Q (x → X) and H : Q → Y. X-systems generalize linear and bilinear systems over a ring. If I and Y are finite, morphic recurrence is necessary and sufficient for a finite realization. For systems over a ring, if X is finite and if Y is an Artinian, injective module, morphic recurrence is necessary and sufficient for an Artinian realization. For systems with equationally-definable state space, a Hankel matrix has finite rank if and only if it has an Artinian, Noetherian realization.