2014/02/28 by Malte Gerhold, Michael Skeide
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Algebra over a field #Cardinal number (linguistics) #Cardinality (data modeling) #Cartesian coordinate system #Cartesian product #Combinatorics #Computer science #Dimension (graph theory) #Discrete mathematics #Factorial #Linguistics #Mathematics #Point (geometry) #Product (mathematics) #Programming language #Pure mathematics #Sequence (biology) #Set (abstract data type) #Word (group theory) #math.CO #math.FA #msc:05A05 #msc:05A15 #msc:46L55 #msc:46L57 #msc:68R15 #semigroups and automata theory
paper · pdf · doi:10.31390/josa.1.4.05
published as Journal of Stochastic Analysis: Vol. 1 : No. 4 , Article 5 (2020) · New title; added references; to appear in Journal of Stochastic Analysis
arxiv created 2020/08/28 · openalex publication_date 2020/11/13 · arxiv updated 2020/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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. Applying this, we provide several new results on the cardinality sequences of word systems and the dimension sequences of subproduct systems.