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A version of o-minimality for the p-adics

  • Deirdre Haskell (a1) and Dugald Macpherson (a2)

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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[2]J. Denef and L. van den Dries , p-adic and real subanalytic sets, Annals of Mathematics, vol. 128 (1988), pp. 79138.

[3]L. van den Dries , Dimension of definable sets, algebraic boundedness and henselian fields, Annals of Pure and Applied Logic, vol. 45 (1989), pp. 189209.

[5]D. Haskell and H. D. Macpherson , Cell decompositions of C-minimal structures, Annals of Pure and Applied Logic, vol. 66 (1994), pp. 113162.

[6]J. Knight , A. Pillay , and C. Steinhorn , Definable sets in ordered structures II, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 593605.

[9]A. J. Macintyre , K. McKenna , and L. van den Dries , Elimination of quantifiers in algebraic structures, Advances in Mathematics, vol. 47 (1983), pp. 7487.

[13]A. Pillay and C. Steinhorn , Definable sets in ordered structures I, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565592.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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