A version of o-minimality for the p-adics

Abstract

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) : x ↦ ax + b, where a ∈ F \ {0} and b ∈ F) 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(y − x) < ν(y − z).

TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

“…Let us now recall some of the properties of this dimension that were already proven by Haskell and Macpherson in [HM97]:…”

“…Hence, theorems (HM 4 ) and (HM 5 ) imply that dim also satisfies the Additivity Property. This fact was not explicitly stated by Haskell and Macpherson in [HM97], and seems to have been somewhat overlooked until now. It plays a crucial role in our proof of Theorem 3.2, hence in all our paper.…”

“…The notion of dimension used by Haskell and Macpherson in [HM97] (which they denoted as topdim A) is defined as follows:…”

“…Inspired by the successes of o-minimality [vdD98] in real algebraic geometry, Haskell and Macpherson [HM97] set out to create a p-adic counterpart, a project which resulted in the notion of P -minimality. One of their achievements was to build a theory of dimension for definable sets which is in many ways similar to the o-minimal case.…”

Cell decomposition and definable functions for weak p‐adic structures

“…An example of this is the concept of o-minimality (cf., e.g., [16]) which inspired Haskell and Macpherson [5] to develop a similar concept, P-minimality, for p-adic fields. A difference between those concepts is that o-minimality also covers reducts of real closed fields R (cf.…”