29 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
The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field
Let R be a sufficiently saturated o-minimal expansion of a real closed field,
let O be the convex hull of the rationals in R, and let st: O^n \to
\mathbb{R}^n be the standard part map. For X \subseteq R^n define st(X):=st(X
\cap O^n). We let \mathbb{R}_{\ind} be the structure with underlying set
\mathbb{R} and expanded by all sets of the form st(X), where X \subseteq R^{n}
is definable in R and n=1,2,.... We show that the subsets of \mathbb{R}^n that
are definable in \mathbb{R}_{\ind} are exactly the finite unions of sets of the
form st(X) \setminus st(Y), where X,Y \subseteq R^n are definable in R. A
consequence of the proof is a partial answer to a question by Hrushovski,
Peterzil and Pillay about the existence of measures with certain invariance
properties on the lattice of bounded definable sets in R^n
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
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
Topological groups, \mu-types and their stabilizers
We consider an arbitrary topological group definable in a structure
, such that some basis for the topology of consists of sets
definable in .
To each such group we associate a compact -space of partial types
which is the quotient of the usual type
space by the relation of two types being "infinitesimally close to
each other". In the o-minimal setting, if is a definable type then it has a
corresponding definable subgroup , which is the stabilizer of
. This group is nontrivial when is unbounded in the sense of
; in fact it is a torsion-free solvable group.
Along the way, we analyze the general construction of and its
connection to the Samuel compactification of topological groups
O-minimal fields with standard part map
Let R be an o-minimal field and V a proper convex subring with residue field
k and standard part (residue) map st: V \to k. Let k_{ind} be the expansion of
k by the standard parts of the definable relations in R. We investigate the
definable sets in k_{ind} and conditions on (R,V) which imply o-minimality of
k_{ind}. We also show that if R is omega-saturated and V is the convex hull of
the rationals in R, then the sets definable in k_{ind} are exactly the standard
parts of the sets definable in (R,V)