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Cell decomposition for semibounded p-adic sets

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Abstract

We study a reduct \({\mathcal{L}_*}\) of the ring language where multiplication is restricted to a neighbourhood of zero. The language is chosen such that for p-adically closed fields K, the \({\mathcal{L}_*}\) -definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K, \({\mathcal{L}_*}\)) can be seen as the p-adic counterpart of the o-minimal structure of semibounded sets. We show that in this language, p-adically closed fields admit cell decomposition, using cells similar to p-adic semi-algebraic cells. From this we can derive quantifier-elimination, and give a characterization of definable functions. In particular, we conclude that multiplication can only be defined on bounded sets, and we consider the existence of definable Skolem functions.

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  1. Department of Mathematics, Purdue University, 150 N. University Street, West-Lafayette, IN, 47907, USA

    Eva Leenknegt

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  1. Eva Leenknegt
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Correspondence to Eva Leenknegt.

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Leenknegt, E. Cell decomposition for semibounded p-adic sets. Arch. Math. Logic 52, 667–688 (2013). https://doi.org/10.1007/s00153-013-0337-8

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  • DOI: https://doi.org/10.1007/s00153-013-0337-8

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