The decomposition of the regular asynchronous systems as parallel connection of regular asynchronous systems

The asynchronous systems are the non-deterministic models of the asynchronous circuits from the digital electrical engineering, where non-determinism is a consequence of the fact that modelling is made in the presence of unknown and variable parameters. Such a system is a multi-valued function f that assigns to an (admissible) input u:R{\to}{0,1}^{m} a set f(u) of (possible) states x:R{\to}{0,1}^{n}. When this assignment is defined by making use of a so-called generator function {\Phi}:{0,1}^{n}{\times}{0,1}^{m}{\to}{0,1}^{n}, then the asynchronous system f is called regular. The generator function {\Phi} acts in this asynchronous framework similarly with the next state function from a synchronous framework. The parallel connection of the asynchronous systems f' and f" is the asynchronous system (f'||f")(u)=f'(u){\times}f"(u). The purpose of the paper is to give the circumstances under which a regular asynchronous system f may be written as a parallel connection of regular asynchronous systems.