1,583 research outputs found
Dimension, matroids, and dense pairs of first-order structures
A structure M is pregeometric if the algebraic closure is a pregeometry in
all M' elementarily equivalent to M. We define a generalisation: structures
with an existential matroid. The main examples are superstable groups of U-rank
a power of omega and d-minimal expansion of fields. Ultraproducts of
pregeometric structures expanding a field, while not pregeometric in general,
do have an unique existential matroid.
Generalising previous results by van den Dries, we define dense elementary
pairs of structures expanding a field and with an existential matroid, and we
show that the corresponding theories have natural completions, whose models
also have a unique existential matroid. We extend the above result to dense
tuples of structures.Comment: Version 2.8. 61 page
Uniform bounds on growth in o-minimal structures
We prove that a function definable with parameters in an o-minimal structure
is bounded away from infinity as its argument goes to infinity by a function
definable without parameters, and that this new function can be chosen
independently of the parameters in the original function. This generalizes a
result in a paper of Friedman and Miller. Moreover, this remains true if the
argument is taken to approach any element of the structure (or plus/minus
infinity), and the function has limit any element of the structure (or
plus/minus infinity).Comment: 3 pages. To appear in Mathematical Logic Quarterl
Towards a Model Theory for Transseries
The differential field of transseries extends the field of real Laurent
series, and occurs in various context: asymptotic expansions, analytic vector
fields, o-minimal structures, to name a few. We give an overview of the
algebraic and model-theoretic aspects of this differential field, and report on
our efforts to understand its first-order theory.Comment: Notre Dame J. Form. Log., to appear; 33 p
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