The lattice structure of behaviors

If a linear, continuous, shift invariant distributed system is considered a s a ( dynamical) system converting input signals to output signals, then th is information is encapsulated in the impulse response or the transfer func tion of the system. The set of all transfer functions has the structure of a ring, corresponding to the operations of parallel and cascade connections of two systems. However, in the behavioral theory of Willems, a system is not described in terms of its input-output transformation property. Indeed, the concept of a behavior does not even need the notions of inputs and out puts and is therefore more fundamental than the classical concept of a syst em given by its transfer function. The question then arises as to what is t he structure of the set of all behaviors. This paper argues that the releva nt structure here is that of a modular lattice.