The notion of independence in categories of algebraic structures. II: S- minimal extensions (Q1111588)

[For part I see ibid. 38, No.2, 185-213 (1988; Zbl 0649.08005).] Let us consider a universal theory T with the amalgamation property in a language consisting uniquely of function symbols, and let \({\mathfrak K}\) be the class of models of T. Let be A,B,H\(\in {\mathfrak K}\), \(A\subset B\subset H\), p a type over A. First, we investigate those extensions of p to B which imply a minimal set of identities with parameters in H. Section 3 deals with the existence and monotonicity-transitivity properties of such extensions. Second, we consider the types q (over some structure F), for which, intuitively, any identity (with parameters in any given extension of F) that q implies is represented in q. We will say that such types are full. Section 4 is concerned with the existence of minimal extensions of a given type, which are full. Actually, we work in a much more general setting than the above. We let L be an arbitrary first-order language and \({\mathfrak K}^ a \)category of L- structures whose morphisms, for simplicity (and until Section 5), will be assumed to be inclusions. It will be apparent in fact that the theory of Sections 3 and 4 goes through in a completely abstract setting (cf. Section 5) that has nothing to do with structures or formulas. Of course, under such general assumptions one can expect to obtain but likewise general results. For instance minimal extensions of types will be shown to exist only as finitely consistent sets of formulas which are therefore not necessarily realized in \({\mathfrak K}\). Nonetheless, such an inquiry enables us to define notions that we believe are useful to investigate in particular classes of structures, such as classes of rings. Anyhow, the results in this paper can be seen as preliminary to Part III where we consider the case of elementary classes.