The mixed product gives a global representation of concurrent systems modelled by interacting automata. In this paper we study the opposite operation: we characterise the transition systems which may be viewed as products and we build some of their decompositions. For a large subclass of systems, we exhibit a minimal decomposition. We finally extend this study to asynchronous automata whose components may be non-deterministic and present an optimal characterisation of the corresponding transition systems. Thus, we state precisely the shape of the transition systems which are associated to three kinds of system; in that way, we obtain axioms which are similar to those identified for the synthesis problem of Petri nets.