Commutative Regular Languages with Product-Form Minimal Automata

We introduce a subclass of the commutative regular languages that is characterized by the property that the state set of the minimal deterministic automaton can be written as a certain Cartesian product. This class behaves much better with respect to the state complexity of the shuffle, for which we find the bound 2nm if the input languages have state complexities n and m, and the upward and downward closure and interior operations, for which we find the bound n. In general, only the bounds (2nm)(vertical bar Sigma vertical bar) and n(vertical bar Sigma vertical bar) are known for these operations in the commutative case. We prove different characterizations of this class and present results to construct languages from this class. Lastly, in a slightly more general setting of partial commutativity, we introduce other, related, language classes and investigate the relations between them.