This paper contributes to a generic theory of behaviour of “finite-state” systems. Systems are coalgebras with a finitely generated carrier for an endofunctor on a locally finitely presentable category. Their behaviour gives rise to the locally finite fixpoint (LFF), a new fixpoint of the endofunctor. The LFF exists provided that the endofunctor is finitary and preserves monomorphisms, is a subcoalgebra of the final coalgebra, i.e. it is fully abstract w.r.t. behavioural equivalence, and it is characterized by two universal properties: as the final locally finitely generated coalgebra, and as the initial fg-iterative algebra. Instances of the LFF are: regular languages, rational streams, rational formal power-series, regular trees etc. Moreover, we obtain e.g. (realtime deterministic resp. non-deterministic) context-free languages, constructively S-algebraic formal power-series (in general, the behaviour of finite coalgebras under the coalgebraic language semantics arising from the generalized powerset construction by Silva, Bonchi, Bonsangue, and Rutten), and the monad of Courcelle's algebraic trees.

本文为“有限状态”系统行为的一般理论做出了贡献。系统是在局部有限可表示类别上具有终结符的有限生成载体的代数。它们的行为引起局部有限的定点（LFF），即endofunctor的新固定点。LFF的存在是前提，即末端终结子是最终的并且保留单态性，是最终子代数的子代数，即它是行为行为上的完全抽象，并且具有两个普遍性质：作为最终的局部有限生成的代数，以及初始fg迭代代数。LFF的实例包括：常规语言，有理流，有理形式化幂级数，有规则树等。此外，我们获得了（实时确定性相对于非确定性）上下文无关的语言，建设性地使用了S。-代数形式幂级数（一般来说，席尔瓦，邦奇，本桑格和鲁滕的广义幂集构造引起的有限阶代数在联合代数语言语义下的行为），以及库尔切勒代数树的单子。