Skip to main content

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
[1]Adeleke, S. and Neumann, P. M., Relations related to betweenness: their structure and automorphisms, Memoirs of the American Mathematical Society, to appear.
[2]Denef, J. and van den Dries, L., p-adic and real subanalytic sets, Annals of Mathematics, vol. 128 (1988), pp. 79138.
[3]van den Dries, L., Dimension of definable sets, algebraic boundedness and henselian fields, Annals of Pure and Applied Logic, vol. 45 (1989), pp. 189209.
[4]van den Dries, L., Haskell, D., and Macpherson, H. D., One-dimensional p-adic subanalytic sets, in preparation.
[5]Haskell, D. and Macpherson, H. D., Cell decompositions of C-minimal structures, Annals of Pure and Applied Logic, vol. 66 (1994), pp. 113162.
[6]Knight, J., Pillay, A., and Steinhorn, C., Definable sets in ordered structures II, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 593605.
[7]Macintyre, A. J., On ω1-categorical theories of fields, Fundamenta Mathematicae, vol. 70 (1971), pp. 125.
[8]Macintyre, A. J., On definable subsets of p-adic fields, this Journal, vol. 41 (1976), pp. 605610.
[9]Macintyre, A. J., McKenna, K., and van den Dries, L., Elimination of quantifiers in algebraic structures, Advances in Mathematics, vol. 47 (1983), pp. 7487.
[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.
[13]Pillay, A. and Steinhorn, C., Definable sets in ordered structures I, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565592.
[14]Prestel, A. and Roquette, P., Formally p-adic fields, Lecture Notes in Mathematics, no. 1050, Springer-Verlag, Berlin, 1984.
[15]Ribenboim, P., Théorie des valuations, Les Presses de l'Université de Montréal, Montréal, 1967.
[16]Scowcroft, P., More on definable sets of p-adic numbers, this Journal, vol. 53 (1988), pp. 912920.
[17]Scowcroft, P. and van den Dries, L., On the structure of semi-algebraic sets over p-adic fields, this Journal, vol. 53 (1988), pp. 11381164.
[18]Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.
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: 6 *
Loading metrics...

Abstract views

Total abstract views: 97 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th March 2018. This data will be updated every 24 hours.