带分解的范畴中的语言与识别
Languages and Recognition in a Category with Factorisation
AI总结：
本文在带分解系统的范畴中，基于纤维化构建代数语言理论的新框架，解决了语言的句法商存在条件及正则语言在J-极限和J-余极限下封闭的条件问题。
AI中文摘要：
通过同态进行的语言识别是代数语言理论的核心构造，最初针对幺半群和半群展开研究，随后被扩展至其他代数结构。本文提出的新范畴化理论基于纤维化，该理论已在自动机理论中得到其他应用。语言与满同态确实会产生两种纤维化，且语言识别的概念在重索引下具有稳定性。我们在带分解系统的范畴中构建该框架，并在纤维化语境下解决两个核心技术问题：其一，给出语言具有句法商（句法同余的推广）的充分条件，并展示如何利用Slomiński的结果在部分具体情形中描述此类商；其二，给出（正则）语言在特定J-极限和J-余极限下封闭的充分条件。
英文摘要：
Language recognition by homomorphisms is a central construction of algebraic language theory. Initially studied for monoids and semigroups, it has subsequently been expanded to other algebraic structures. Our new categorical account is based on fibrations, which have already seen other applications in automata theory. Languages and surjective homomorphisms give indeed rise to two fibrations, and the notion of language recognition is stable under reindexing. We develop this framework in a category with a factorisation system and address two main technical questions in the fibrational setting. First, we provide sufficient conditions under which languages have syntactic quotients (which is a generalisation of syntactic congruences) and we show how such quotients can be described in some concrete cases using a result by Slomiński. Second, we introduce sufficient conditions under which (regular) languages are closed under certain J-limits and J-colimits.
