this paper, so we leave them out here. Thus we can apply the idea of combining things via colimits to institutions themselves, with the special point that we have to take limits here instead of colimits. Taking limits in CAT results in categories of "amalgamated objects", i. e. we put signatures and models together at the level of single objects. In contrast to this, sentences are combined with colimits in Set (due to the contravariant direction of the sentence component). That is, sets of sentences are combined. To show how this works, we introduce some well-known institutions and morphisms between them.

In this article we present a functional specification language based on inclusions between set expressions. Instead of computing with data individuals we deal with their classification into sets. The specification of functions and relations by means of inclusions can be considered as a generalization of the conventional algebraic specification by means of equations. The main aim of this generalization is to facilitate the incremental refinement of specifications. Furthermore, inclusional specifications admit a natural visual syntax which can also be used to visualize the reasoning process. We show that reasoning with inclusions is well captured by bi-rewriting, a rewriting technique introduced by Levy and Agust'i [15]. However, there are still key problems to be solved in order to have executable inclusional specifications, necessary for rapid prototyping purposes. The article mainly points to the potentialities and difficulties of specifying with inclusions.

The thesis of this paper is simple: that understanding both the construction and the interpretation of algebras is useful not only as a specification technique for the implementation of interactive systems, but also as providing fundamental insights into the design and use of such systems in a general way. This is in contrast firstly to state-oriented specification techniques like Z and VDM , which are less mathematically tractable and so obscure the general observations we wish to make here. It is in contrast secondly to behavioural notations like CSP and CCS which, while very tractable, abstract away from issues of value or state and only describe permissible action orderings. The paper presents a generic algebra for direct manipulation interactive systems, and indicates the importance of initial and final interpretations of algebraic specifications in the iterative design process. 1 Introduction We start with the recognition that if we want to describe or specify the functional...