n-ary Cartesian composition of automata

In our paper, we construct Cartesian composition of automata in a way rather different from the classical approach. In our case, the resulting structure is not an automaton but a quasi-multiautomaton, i.e., a structure whose input alphabet is a semi-hypergroup instead of a set or a free monoid. In our reasoning, we make use of earlier results on complete (semi)hypergroups and show that this approach not only simplifies our construction but also yields some natural applications.