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# A version of o-minimality for the p-adics

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In this paper we formulate a notion similar to o-minimality but appropriate for the p-adics. The paper is in a sense a sequel to [11] and [5]. In [11] a notion of minimality was formulated, as follows. Suppose that L, L+ are first-order languages and + is an L+-structure whose reduct to L is . Then + is said to be -minimal if, for every N+ elementarily equivalent to +, every parameterdefinable subset of its domain N+ is definable with parameters by a quantifier-free L-formula. Observe that if L has a single binary relation which in is interpreted by a total order on M, then we have just the notion of strong o-minimality, from [13]; and by a theorem from [6], strong o-minimality is equivalent to o-minimality. If L has no relations, functions, or constants (other than equality) then the notion is just strong minimality.

In [11], -minimality is investigated for a number of structures . In particular, the C-relation of [1] was considered, in place of the total order in the definition of strong o-minimality. The C-relation is essentially the ternary relation which naturally holds on the maximal chains of a sufficiently nice tree; see [1], [11] or [5] for more detail, and for axioms. Much of the motivation came from the observation that a C-relation on a field F which is preserved by the affine group AGL(1,F) (consisting of permutations (a,b) : xax + b, where aF \ {0} and bF) is the same as a non-trivial valuation: to get a C-relation from a valuation ν, put C(x;y,z) if and only if ν(yx) < ν(yz).

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[10]Macpherson, H. D., Marker, D., and Steinhorn, C., Weakly o-minimal structures and real closed fields, in preparation.
[11]Macpherson, H. D. and Steinhorn, C., On variants of o-minimality, Annals of Pure and Applied Logic, to appear.
[12]Mathews, L., The independence property in unstable algebraic structures I: p-adically closed fields, preprint.
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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
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