A version of o-minimality for the p-adics

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 ).