From T-Coalgebras to Filter Structures and Transition Systems
نویسنده
چکیده
For any set-endofunctor T : Set→ Set there exists a largest subcartesian transformation μ to the filter functor F : Set → Set. Thus we can associate with every T -coalgebra A a certain filter-coalgebra AF. Precisely, when T weakly preserves preimages, μ is natural, and when T weakly preserves intersections, μ factors through the covariant powerset functor P, thus providing for every T -coalgebra A a Kripke structure AP. The paper characterizes weak preservation of preimages, of intersections, and preservation of both preimages and intersections by a functor T via the existence of transformations from T to either F or P. Moreover, we define for arbitrary T -coalgebras A a next-time operator©A with associated modal operators 2 and 3 and relate their properties to weak limit preservation properties of T . In particular, for any T -coalgebra A there is a transition system K with ©A = ©K if and only if T weakly preserves intersections.
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