Coalgebras are of growing importance in theoretical computer science. To develop languages for them is significant for the specification and verification of systems modelled with them. Modal logic has proved to be suitable for this purpose. So far, most approaches have presented a language to describe only deterministic coalgebras. The present paper introduces a generalization that also covers non-deterministic systems. As a special case, we obtain the "usual" modal logic for Kripke-structures. Models for our modal language L F are F-coalgebras where the functor F is inductively constructed from constant sets and the identity functor using product, coproduct, exponentiation, and the power set functor. We define a language L F and show that it embeds into L F . We prove that, for image-finite coalgebras, L F is expressive enough to distinguish elements up to bisimilarity and therefore L F does so, too. Moreover, we also give a complete calculus for L F in case the constants...

. Data Structures arising in programming are conveniently modeled by universal algebras. State based and object oriented systems may be described in the same way, but this requires that the state is explicitly modeled as a sort. From the viewpoint of the programmer, however, it is usually intended that the state should be "hidden" with only certain features accessible through attributes and methods. States should become equal, if no external observation may distinguish them. It has recently been discovered that state based systems such as transition systems, automata, lazy data structures and objects give rise to structures dual to universal algebra, which are called coalgebras. Equality is replaced by indistinguishability and co-induction replaces induction as proof principle. However, as it turns out, one has to look at universal algebra from a more general perspective (using elementary category theoretic notions) before the dual concept is able to capture the relevant ...

. Functors preserving weak pullbacks provide the basis for a rich structure theory of coalgebras. We give an easy to use criterion to check whether a functor preserves weak pullbacks. We apply the characterization to the functor F which associates a set X with the set F(X) of all filters on X. It turns out that this functor preserves weak pullbacks, yet does not preserve weak generalized pullbacks. Since topological spaces can be considered as F- coalgebras, in fact they constitute a covariety, we find that the intersection of subcoalgebras need not be a coalgebra, and 1-generated F-coalgebras need not exist. 1. Introduction Coalgebras have been introduced by Aczel and Mendler [AM89] to model various types of transition systems. Reichel [Rei95], and Jacobs [Jac96] show that coalgebras are well suited for modeling object oriented programmming and for program verification. In [Rut96], J.J.M.M. Rutten develops the a fundamental theory of "universal coalgebra" along the lines of univers...