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Extensional quotient coalgebras. (English) Zbl 1390.68482

Summary: Given an endofunctor \(\mathsf{F}\) of an arbitrary category, any maximal element of the lattice of congruence relations on an \(\mathsf{F}\)-coalgebra \((\mathsf{A}, a)\) is called a coatomic congruence relation on \((\mathsf{A}, a)\). Besides, a coatomic congruence relation \(\mathsf{K}\) is said to be factor split if the canonical homomorphism \(\nu : \mathsf{A}_{\mathsf{K}} \rightarrow \mathsf{A}_{\nabla_{\mathsf{A}}}\) splits, where \(\nabla_{\mathsf{A}}\) is the largest congruence relation on \((\mathsf{A}, a)\). Assuming that \(\mathsf{F}\) is a covarietor which preserves regular monos, we prove under suitable assumptions on the underlying category that, every quotient coalgebra can be made extensional by taking the regular quotient of an \(\mathsf{F}\)-coalgebra with respect to a coatomic and not factor split congruence relation or its largest congruence relation.

MSC:

68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
18C50 Categorical semantics of formal languages
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