Subproduct systems and Cartesian systems; new results on factorial languages and their relations with other areas

Understand

We point out that a sequence of natural numbers is the dimension sequence of a subproduct system if and only if it is the cardinality sequence of a word system (or factorial language).

• Determining such sequences is, therefore, reduced to a purely combinatorial problem in the combinatorics of words.
• A corresponding (and equivalent) result for graded algebras has been known in abstract algebra, but this connection with pure combinatorics has not yet been noticed by the product systems community.
• We also introduce Cartesian systems, which can be seen either as a set theoretic version of subproduct systems or an abstract version of word systems.