Strongly determined types Original Research Article

The notion of a strongly determined type over A extending p is introduced, where p .∈ S(A). A strongly determined extension of p over A assigns, for any model M ⊂)- A, a type q ∈ S(M) extending p such that, if View the MathML source realises q, then any elementary partial map M → M which fixes acleq(A) pointwise is elementary over View the MathML source. 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.