Applied Categorical Structures

, Volume 16, Issue 3, pp 313–332 | Cite as

On Minimal Coalgebras

  • H. Peter GummEmail author


We define an out-degree for F-coalgebras and show that the coalgebras of outdegree at most κ form a covariety. As a subcategory of all F-coalgebras, this class has a terminal object, which for many problems can stand in for the terminal F-coalgebra, which need not exist in general. As examples, we derive structure theoretic results about minimal coalgebras, showing that, for instance minimization of coalgebras is functorial, that products of finitely many minimal coalgebras exist and are given by their largest common subcoalgebra, that minimal subcoalgebras have no inner endomorphisms and show how minimal subcoalgebras can be constructed from Moore-automata. Since the elements of minimal subcoalgebras must correspond uniquely to the formulae of any logic characterizing observational equivalence, we give in the last section a straightforward and self-contained account of the coalgebraic logic of D. Pattinson and L. Schröder, which we believe is simpler and more direct than the original exposition.


Coalgebra minimal coalgebra Moore-automata Products Modal logic Coalgebraic logic 

Mathematics Subject Classifications (2000)

18B20 68Q85 16W30 


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© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany

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