Strongly determined types

The notion of a strongly determined type over A extending p is introduced, where p is an element of S(A). A strongly determined extension of p over A assigns, for any model M superset of or equal to A, a type q is an element of S(M) extending p such that, if (c) over bar realises q, then any elementary partial map M --> M which fixes acl(eq)(A) pointwise is elementary over (c) over bar. This gives a crude notion of independence (over A) which arises very frequently. Examples are provided of many different kinds of theories with strongly determined types, and some without. We investigate a notion of multiplicity for strongly determined types with applications to 'involved' finite simple groups, and an analogue of the Finite Equivalence Relation Theorem. Lifting of strongly determined types to covers of a structure (and to symmetric extensions) is discussed, and an application to finite covers is given. (C) 1999 Elsevier Science B.V. All rights reserved.