48 research outputs found
Dependent Pairs
We prove that certain pairs of ordered structures are dependent. Among these
structures are dense and tame pairs of o-minimal structures and further the
real field with a multiplicative subgroup with the Mann property, regardless of
whether it is dense or discrete
Definable choice for a class of weakly o-minimal theories
Given an o-minimal structure with a group operation, we show
that for a properly convex subset , the theory of the expanded structure
has definable Skolem functions precisely when
is valuational. As a corollary, we get an elementary proof that
the theory of any such does not satisfy definable choice.Comment: 11 page
Stratifications of tangent cones in real closed (valued) fields
We introduce tangent cones of subsets of cartesian powers of a real closed
field, generalising the notion of the classical tangent cones of subsets of
Euclidean space. We then study the impact of non-archimedean stratifications
(t-stratifications) on these tangent cones. Our main result is that a
t-stratification induces stratifications of the same nature on the tangent
cones of a definable set. As a consequence, we show that the archimedean
counterpart of a t-stratification induces Whitney stratifications on the
tangent cones of a semi-algebraic set. The latter statement is achieved by
working with the natural valuative structure of non-standard models of the real
field.Comment: 15 page
The elementary theory of Dedekind cuts in polynomially bounded structures
Let M be a polynomially bounded, o-minimal structure with archimedean prime
model, for example if M is a real closed field. Let C be a convex and unbounded
subset of M. We determine the first order theory of the structure M expanded by
the set C. We do this also over any given set of parameters from M, which
yields a description of all subsets of M^n, definable in the expanded
structure.Comment: 16 pages. The paper is a sequel to
http://www-nw.uni-regensburg.de/~.trm22116.mathematik.uni-regensburg.de/paper
s/cutsa.p
Model completeness of o-minimal fields with convex valuations
We let R be an o-minimal expansion of a field, V a convex subring, and an elementary substructure of (R,V). We let L be the language
consisting of a language for R, in which R has elimination of quantifiers, and
a predicate for V, and we let be the language L expanded by
constants for all elements of . Our main result is that (R,V) considered
as an -structure is model complete provided that , the
corresponding residue field with structure induced from R, is o-minimal. Along
the way we show that o-minimality of implies that the sets definable in
are the same as the sets definable in k with structure induced from
(R,V). We also give a criterion for a superstructure of (R,V) being an
elementary extension of (R,V)
The exponential rank of non-Archimedean exponential fields
Based on the work of Hahn, Baer, Ostrowski, Krull, Kaplansky and the
Artin-Schreier theory, and stimulated by a paper of S. Lang in 1953, the theory
of real places and convex valuations has witnessed a remarkable development and
has become a basic tool in the theory of ordered fields and real algebraic
geometry. In this paper, we take a further step by adding an exponential
function to the ordered field
Distal and Non-Distal Pairs
The aim of this note is to determine whether certain non-o-minimal expansions
of o-minimal theories which are known to be NIP, are also distal. We observe
that while tame pairs of o-minimal structures and the real field with a
discrete multiplicative subgroup have distal theories, dense pairs of o-minimal
structures and related examples do not
O-minimal spectrum
Let X be a definable sub-set of some o-minimal structure. We study the
spectrum of X, in relation with the definability of types
Real closed valued fields with analytic structure
We show quantifier elimination theorems for real closed valued fields with
separated analytic structure and overconvergent analytic structure in their
natural one-sorted languages and deduce that such structures are weakly
o-minimal. We also provide a short proof that algebraically closed valued
fields with separated analytic structure (in any rank) are -minimal.Comment: 10 pages. Any comments welcome
O-minimal residue fields of o-minimal fields
Let R be an o-minimal field with a proper convex subring V. We axiomatize the
class of all structures (R,V) such that k_ind, the corresponding residue field
with structure induced from R via the residue map, is o-minimal. More
precisely, in previous work it was shown that certain first order conditions on
(R,V) are sufficient for the o-minimality of k_ind. Here we prove that these
conditions are also necessary