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Aleph-zero-categoricity over a predicate. (English) Zbl 0666.03028

Tübingen: Univ. Tübingen, Math. Fakultät, 57 S. (1988).
Categoricity over a predicate is defined in the following situation. There are two languages L and \(L_ 0\) such that \(L_ 0\) is a sublanguage of L and L contains a unary predicate P. The languages are considered to be relational, so in every L-structure the set of elements satisfying the predicate P is in a natural way an \(L_ 0\)-structure, which is called the P-part of the original L-structure....
Note that if the P-part is empty, then a theory can be strongly categorical over a predicate only when it is the theory of a finite structure. Taking note of this fact, A. Pillay [Notre Dame J. Formal Logic 24, 527-536 (1983; Zbl 0491.03009)] introduced the concept of \(\aleph_ 0\)-categoricity over \((P,L_ 0)\), i.e. if two countable models have the same P-part, then there is an L-isomorphism between these models which is the identity on the P-part. Now if the P-part is empty this last concept reduces to \(\aleph_ 0\)-categoricity and if the P-part equals the whole model it reduces to implicit definability....
In this thesis I restrict myself to \(\aleph_ 0\)-categoricity over \((P,L_ 0)\). The first chapter is concerned with the general case of an \(\aleph_ 0\)-categorical theory over \((P,L_ 0)....\)
Chapter two is concerned only with \(\aleph_ 0\)-categoricity over \((P,L_ 0)\) for Abelian groups with a distinguished subgroup.

MSC:

03C35 Categoricity and completeness of theories
20A15 Applications of logic to group theory