On the serial connection of the regular asynchronous systems

The asynchronous systems f are multi-valued functions, representing the non-deterministic models of the asynchronous circuits from the digital electrical engineering. In real time, they map an ’admissible input’ function u : R → {0, 1}m to a set f (u) of ’possible states’ x ∈ f (u), where x : R → {0, 1}n. When f is defined by making use of a ’generator function’ Φ : {0, 1}n × {0, 1}m → {0, 1}n, the system is called regular. The usual definition of the serial connection of systems as composition of multi-valued functions does not bring the regular systems into regular systems, thus the first issue in this study is to modify in an acceptable manner the definition of the serial connection in a way that matches regularity. This intention was expressed for the first time, without proving the regularity of the serial connection of systems, in the work [3]. Our present purpose is to restate with certain corrections and prove Theorem 45 from that work.