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Let G be a model of Presburger arithmetic. Let be an expansion of the language of Presburger . In this paper, we prove that the ‐theory of G is ‐minimal iff it has the exchange property and is definably complete (i.e., any bounded definable set has a maximum). If the ‐theory of G has the exchange property but is not definably complete, there is a proper definable convex subgroup H. Assuming...
Let be the field of p‐adic numbers in the language of rings. In this paper we consider the theory of expanded by two predicates interpreted by multiplicative subgroups and where are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if α and β have positive p‐adic valuation. If either α or β has zero valuation we show that the theory...
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