A version of o-minimality for the p-adics Academic Article uri icon

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abstract

  • In this paper we formulate a notion similar too-minimality but appropriate for thep-adics. The paper is in a sense a sequel to [11] and [5]. In [11] a notion of minimality was formulated, as follows. Suppose thatL, L+are first-order languages and+is anL+-structure whose reduct toLis. Then+is said to be-minimal if, for everyN+elementarily equivalent to+, every parameterdefinable subset of its domainN+is definable with parameters by a quantifier-freeL-formula. Observe that ifLhas a single binary relation which inis interpreted by a total order onM, then we have just the notion ofstrong o-minimality, from [13]; and by a theorem from [6], strongo-minimality is equivalent too-minimality. IfLhas no relations, functions, or constants (other than equality) then the notion is juststrong minimality.In [11],-minimality is investigated for a number of structures. In particular, theC-relationof [1] was considered, in place of the total order in the definition of strongo-minimality. TheC-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 aC-relation on a fieldFwhich is preserved by the affine group AGL(1,F) (consisting of permutations (a,b) :xax+b, whereaF\ {0} andbF) is the same as a non-trivial valuation: to get aC-relation from a valuation ν, putC(x;y,z) if and only if ν(yx) < ν(yz).

publication date

  • December 1997