Skip to main content
    • Aa
    • Aa

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).

Hide All
[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 80 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th October 2017. This data will be updated every 24 hours.