The intersection and the union of the asynchronous systems

The asynchronous systems $f$ are the models of the asynchronous circuits from digital electrical engineering. They are multi-valued functions that associate to each input $u:\mathbf{R}\to \{0,1\}^{m}$ a set of states $x\in f(u),$ where $x:\mathbf{R}\to \{0,1\}^{n}.$ The intersection of the systems allows adding supplementary conditions in modeling and the union of the systems allows considering the validity of one of two systems in modeling, for example when testing the asynchronous circuits and the circuit is supposed to be 'good' or 'bad'. The purpose of the paper is that of analyzing the intersection and the union against the initial/final states, initial/final time, initial/final state functions, subsystems, dual systems, inverse systems, Cartesian product of systems, parallel connection and serial connection of systems.