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Preprint Number 275

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275. Wilfrid Hodges
Proof of Gaifman's conjecture for relatively categorical abelian groups

Submission date: 17 October 2010.

Abstract:

Haim Gaifman conjectured in 1974 that if T is a complete first-order theory which is relatively categorical over its relativisation T^P to a predicate P, then every model B of T^P can be extended to a model A of T with A^P = B. He proved this when T is rigid over P, and it holds also for some cases of relative categoricity in uncountably categorical theories. In what may be the first example going significantly beyond these two types, we show that the conjecture is true when the relatively categorical theory T is the theory of an abelian group with P selecting a subgroup.

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