Cartesian composition and the problem of generalizing the MAC condition to quasi-multiautomata

When we assume that the input-set of an automaton without output is a semihypergroup instead of a monoid, we talk about quasi-multiautomata. Even though cartesian composition of quasi-automata is a commonly used concept, the cartesian composition of quasi-multiautomata has not been successfully constructed yet. In our paper we show that the straightforward transfer of the definition into the multivariate context fails. We suggest two possible solutions of this problem.