. 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 ...

Given a # -complete (semi)lattice L, we consider L-labeled transition systems as coalgebras of a functor L (-) , associating with a set X the set L X of all L-fuzzy subsets. We describe simulations and bisimulations of L-coalgebras to show that L (-) weakly preserves nonempty kernel pairs i# it weakly preserves nonempty pullbacks i# L is join infinitely distributive (JID). Exchanging L for a commutative monoid M, we consider the functor M (-) # which associates with a set X all finite multisets containing elements of X with multiplicities m # M . The corresponding functor weakly preserves nonempty pullbacks along injectives i# 0 is the only invertible element of M, and it preserves nonempty kernel pairs i# M is refinable, in the sense that two sum representations of the same value, r 1 + . . . + r m = c 1 + . . . + c n , have a common refinement matrix (m i,j ) whose k-th row sums to r k and whose l-th column sums to c l for any 1 # k # m and 1 # l # n.

We prove that for any F the category of F -coalgebras is distributive if F preserves preimages, i.e. pullbacks along an injective map, and that the converse is also true whenever has finite products.

The existence of a final coalgebra is equivalent to the existence of a formal logic with a set (small class) of formulas that has the Hennessy-Milner property of distinguishing coalgebraic states up to bisimilarity. This applies to coalgebras of any functor on the category of sets for which the bisimilarity relation is transitive. There are cases of functors that do have logics with the Hennessy-Milner property, but the only such logics have a proper class of formulas. The main theorem gives a representation of states of the final coalgebra as certain satisfiable sets of formulas. The key technical fact used is that any function between coalgebras that is truth-preserving and has a simple codomain must be a coalgebraic morphism.

Universal coalgebra is a mathematical theory of state based systems, which in many respects is dual to universal algebra. Equality must be replaced by indistinguishability. Coinduction replaces induction as a proof principle and maps are defined by co-recursion. In this (entirely self-contained) paper we give a first glimpse at the general theory and focus on some applications in Computer Science. 1. State based systems State based systems can be found everywhere in our environment -- from simple appliances like alarm clocks and answering machines to sophisticated computing devices. Typically, such systems receive some input and, as a result, produce some output. In contrast to purely algebraic systems, however, the output is not only determined by the input received, but also by some modifiable "internal state". Internal states are usually not directly observable, so there may as well be di#erent states that cannot be distinguished from the input-output behavior of the system. A simple example of a state based system is a digital watch with several buttons and a display. Clearly, the buttons that are pressed do not by themselves determine the output - it also depends on the internal state, which might include the current time, the mode (time/alarm/stopwatch), and perhaps the information which buttons have been pressed previously. The user of a system is normally not interested in knowing precisely, what the internal states of the system are, nor how they are represented. Of course, he might try to infer all possible states by testing various input-output combinations and attribute di#erent behaviors to di#erent states. Some states might not be distinguishable by their outside behavior. It is therefore natural to define an appropriate indistinguishability relation "#...

Implications in a category can be presented as epimorphisms: an object satisfies the implication iff it is injective w.r.t. that epimorphism. G. Roçu formulated a logic for deriving an implication from other implications. We present two versions of implicational logics: a general one and a finitary one (for epimorphisms with finitely presentable domains and codomains). In categories Alg Σ of algebras on a given signature our logic specializes to the implicational logic of R. Quackenbush. In categories Coalg H of coalgebras for a given accessible endofunctor H of sets we derive a logic for implications in the sense of P. Gumm. 1

ABSTRACT. We survey coalgebras as models of state based systems together with their global and local logics. We convey some useful intuition regarding Set-functors which leads naturally to coalgebraic modal logic where modalities are validity patterns for the successor object of a state. 1.