Abstract
In this paper, we consider reducts of p-adically closed fields. We introduce a notion of shadows: sets \({M_f = \{(x,y) \in K^2 \mid |y| = |f(x)|\}}\) , where f is a semi-algebraic function. Adding symbols for such sets to a reduct of the ring language, we obtain expansions of the semi-affine language where multiplication is nowhere definable, thus giving a negative answer to a question posed by Marker, Peterzil and Pillay. The second main result of this paper is the fact that in p-adic fields, full multiplication becomes definable if we add a rational function to the semi-affine language.
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Leenknegt, E. Reducts of p-adically closed fields. Arch. Math. Logic 53, 285–306 (2014). https://doi.org/10.1007/s00153-013-0366-3
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DOI: https://doi.org/10.1007/s00153-013-0366-3
Keywords
- Cell decomposition
- Quantifier elimination
- p-adic numbers
- P-minimality
- o-minimality
- Reducts
- Defining multiplication
- Skolem functions