Archive for Mathematical Logic

, Volume 52, Issue 5–6, pp 667–688 | Cite as

Cell decomposition for semibounded p-adic sets



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.


Cell decomposition Quantifier elimination p-adic numbers p-minimality o-minimality Definability Restricted multiplication 

Mathematics Subject Classification (2000)

03C07 03C10 03C64 11U09 12J12 


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest-LafayetteUSA

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