子积系统与笛卡尔系统：阶乘语言的新结果及其与其它领域的关系
Subproduct systems and Cartesian systems; new results on factorial languages and their relations with other areas
摘要：
指出一个自然数序列是一个子积系统的维数序列当且仅当它是一个词系统（或阶乘语言）的基数序列。因此，确定这样的序列被简化为单词组合学中的纯组合问题。分次代数的一个相应的（和等价的）结果在抽象代数中是已知的，但是这种与纯组合学的联系还没有被乘积系统社区注意到。我们还介绍笛卡尔系统，它可以被看作是一个集理论版本的子产品系统或一个抽象版本的字系统。应用这一结果，我们给出了关于词系统的基数序列和子积系统的维数序列的几个新结果。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
