We show the interest of the mixed (or synchronous) product for studying traces and trace languages in the free partially commutative monoids. We use this product to construct particular asynchronous automata and we show that each asynchronous automaton is the image by a strictly alphabetic morphism of a mixed product of automata. Then we use this result and Zielonka's theorem (1984) to obtain a result similar to Mezei's theorem in the free partially commutative monoids by showing that each recognizable trace language is the homomorphic image of some finite union of synchronous products of recognizable languages.