This chapter is concerned with the relation between types or, more properly, pp-types, and the structure of their hulls. Some of the results proved in this chapter will be used in the more global considerations of Chapter 10 but, in the main, we concentrate here on local structure.

Replacement of minimal pp-definable subgroups by minimal pairs is a key step in going from the totally transcendental to the general case. Irreducible types which share a minimal pair need not be equal, but they do have isomorphic hulls (§1). In consequence, given a complete theory, each unlimited indecomposable pure-injective has the same multiplicity in every pureinjective model. Another result which was seen in the totally transcendental case in §4.6 and is now proved in full generality, is that if an indecomposable pure-injective has a minimal pair, then the corresponding quotient of subgroups has the structure of a 1-dimensional vectorspace over the division ring associated to the indecomposable (9.6).

If two types have linked realisations then they have isomorphic parts: specifically, between the positive and negative parts of each type there is an interval, defined in terms of the linking formula, and these intervals are isomorphic (with positive and negative parts corresponding). This means that irreducible types which have isomorphic hulls are syntactically similar. For instance, if one of them has one of the (syntactically defined) dimensions of Chapter 10, then so does the other (and the values are equal). Finally in §2, the relation between the hull of a type and its direct summands is explicated (9.16) and there is a syntactic criterion on the type for there to be an indecomposable direct summand of its hull.