On Minimal Coalgebras

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. For every automaton A there exists a minimal automaton ∇(A), which displays the same behavior as A. In the case of acceptors, this means that A and ∇(A) recognize the same language, and in the more general case of Moore-Automata it means that equal inputs generate the same outputs. Minimality, of course, refers to the cardinality of (the state set of) any automaton displaying the same behavior. Turning to coalgebras, there are two possible notions of state equivalence, to begin with. Given two coalgebras A and B one may consider states a ∈ A and b ∈ B equivalent if they are bisimilar, or, alternatively, if they are observationally equivalent. Whereas these two notions agree for automata, they may differ for general coalgebras, unless the type functor F weakly preserves kernels, see [7]. In general, bisimilar states are observationally equivalent, but the converse need not hold. Observational equivalence, restricted to a single coalgebra, is a congruence relation, i.e. the kernel of some homomorphism, while bisimilarity may fail to be transitive. For these reasons we choose observational equivalence as our notion of equivalence. It follows that for every coalgebra A there exists an equivalent minimal coalgebra ∇(A). Now A and B are observationally equivalent just in case ∇(A) and ∇(B) are isomorphic. Generalizing a result of Kianpi and Jugnia [10], we show that the class of all minimal coalgebras forms a full subcategory of the category of all F -coalgebras and ∇ is a functor, which is left adjoint to the inclusion functor. This and similar results are most easily obtained when a terminal F -coalgebra T exists. In this case ∇(A) is isomorphic to the image of A under the unique homomorphism τA : A → T and arbitrary coalgebras A and B are equivalent iff their terminal images are identical, i.e. τA[A] = τB[B]. From this many results follow easily that, for instance, • products of finitely many minimal coalgebras exist, • the structure maps of minimal coalgebras must be injective,