5 research outputs found
Quantifier elimination in ordered abelian groups
We give a new proof of quantifier elimination in the theory of all ordered
abelian groups in a suitable language. More precisely, this is only "quantifier
elimination relative to ordered sets" in the following sense. Each definable
set in the group is a union of a family of quantifier free definable sets,
where the parameter of the family runs over a set definable (with quantifiers)
in a sort which carries the structure of an ordered set with some additional
unary predicates.
As a corollary, we find that all definable functions in ordered abelian
groups are piecewise affine linear on finitely many definable pieces.Comment: 30 page
Vapnik-Chervonenkis density in some theories without the independence property, I
We recast the problem of calculating Vapnik-Chervonenkis (VC) density into
one of counting types, and thereby calculate bounds (often optimal) on the VC
density for some weakly o-minimal, weakly quasi-o-minimal, and -minimal
theories.Comment: 59