This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributive laws. 1

It is well known that type constructors of incomplete trees (trees with variables) carry the structure of a monad with substitution as the extension operation. Less known are the facts that the same is true of type constructors of incomplete cotrees (=non-wellfounded trees) and that the corresponding monads exhibit a special structure. We wish to draw attention to the dual facts which are as meaningful for functional programming: type constructors of decorated cotrees carry the structure of a comonad with redecoration as the coextension operation, and so do---even more interestingly---type constructors of decorated trees.

This paper introduces coalgebraic monads as a unified model of term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary coalgebraic monads which allows us to generalise the notion of rational term and term graph to categories other than Set. As an application we sketch part of the correctness of the the term graph implementation of functional programming languages.

this paper express that the above principles work under di#erent additional assumptions which are needed to show that the large system can actually be constructed inside the category. The basic Theorem requires the existence of countable coproducts. Later we also present a variant where the functor T comes a as a monad, the functor F is taken from a copointed functor, and the distributive law # is assumed to interact nicely with this additional structure (i.e. # should be a distributive law of the monad over the copointed functor, see again (Lenisa et al., 2000))