On the lattice of behaviors

The behavioral approach to systems and control, developed by J. C. Willems, provides a conceptually attractive framework for mathematical systems theory. An important result of U. Oberst implies that for certain signal spaces, the set of all linear behaviors contained in that signal space has the structure of a lattice, the two relevant operations being summation and intersection. This has been noted by S. Shankar who has investigated some fundamental properties of this lattice. In the present paper, we continue this line of research and study lattice-theoretic concepts such as atoms, reducibility, and complements, in the context of multidimensional behaviors. In particular, the notion of complementation plays an important role for the control of multidimensional systems.