The main features of finite multidimensional behaviors are introduced as properties of the trajectories supports and connected with the polynomial matrices adopted for their description.Observability and local detectability are shown to be equivalent to the kernel representation of a behavior via some parity check matrix H-T. The main properties of locally undetectable behaviors as well as their connections with the notion of constrained variables are investigated, and a general representation result for finite support behaviors is derived.The input-output representation via generator matrices is finally discussed, and some connections between matrix primeness and the constraints every trajectory imposes on the support of the corresponding input are analyzed.