The notion of dependence (forking) considered in the last chapter is the raw stuff of stability theory: for the analyses of models, we need some tools shaped from it. For instance, vectorspaces over a field are “unidimensional” in the sense that only one parameter (the dimension over the field) is needed to classify them. But, for classifying abelian groups of exponent 4, we need two “dimensions” (the number of copies of ℤ2 and the number of copies of ℤ4). These “dimensions” (unfortunately, the term is overworked) are exposed by using notions such as orthogonality and weight. The chapter is concerned with this “structural” level of stability theory.

Again, I have tried to include enough explanation for non-specialists, since some of the results are used elsewhere, and since an understanding of the main points adds another dimension to one's view of later results.

The first task is to extract that part of a type, its “free part”, which controls the stabilitytheoretic properties of the type. This is done in section 1.

An element is said to dominate another if, whenever a third element is independent of the first, then it is independent of the second also. For instance, an element dominates all elements in any copy of its hull (at least, that is so for unlimited types). The exact connection between domination and hulls is clarified in §2. A type is RK-minimal if, whenever a realisation of it dominates another element, the latter also dominates the first. Since domination corresponds to realisation in prime pure-injective extensions, such RK-minimal types correspond to the simplest building blocks of models. Indecomposable pure-injectives are among these simplest blocks, but there can be others.