Abstract This note introduces practical set invariance notions for physically interconnected, discrete–time systems, subject to additive but bounded disturbances. The developed approach provides a decentralized, non–conservative and computationally tractable way to study desirable robust positive invariance and stability notions for the overall system as well as to guarantee safe and independent operation of the constituting subsystems. These desirable properties are inherited, under mild assumptions, from the classical stability and invariance properties of the associated vector–valued dynamics which capture in a simple but appropriate and non–conservative way the dynamical behavior induced by the underlying set–dynamics of interest.