Submission date: 21 January 2016.

Abstract:

Mathematics Subject Classification: We show that if G is a stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then G is superstable of finite U-rank. Combined with recent work of Palacin and Sklinos, we conclude that ( Z , + , 0 ) has no proper stable expansions of finite weight. A corollary of this result is that if P ⊆ Z^n is definable in a finite dp-rank expansion of ( Z , + , 0 ), and ( Z , + , 0 , P ) is stable, then P is definable in ( Z , + , 0 ). In particular, this answers a question of Marker on stable expansions of the group of integers by sets definable in Presburger arithmetic.

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Full text arXiv 1601.05692: pdf, ps.