771 research outputs found
Dp-minimality: basic facts and examples
We study the notion of dp-minimality, beginning by providing several
essential facts, establishing several equivalent definitions, and comparing
dp-minimality to other minimality notions. The rest of the paper is dedicated
to examples. We establish via a simple proof that any weakly o-minimal theory
is dp-minimal and then give an example of a weakly o-minimal group not obtained
by adding traces of externally definable sets. Next we give an example of a
divisible ordered Abelian group which is dp-minimal and not weakly o-minimal.
Finally we establish that the field of p-adic numbers is dp-minimal.Comment: 19 pages; simplified proof for the p-adic
Left-ordered inp-minimal groups
We prove that any left-ordered inp-minimal group is abelian, and we provide
an example of a non-abelian left-ordered group of dp-rank 2
On uniform definability of types over finite sets
In this paper, using definability of types over indiscernible sequences as a
template, we study a property of formulas and theories called "uniform
definability of types over finite sets" (UDTFS). We explore UDTFS and show how
it relates to well-known properties in model theory. We recall that stable
theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence.
We then show that all dp-minimal theories have UDTFS.Comment: 17 pages, 0 figure
Witnessing dp-rank
We prove that in NTP_2 theories if p is a dependent type with dp-rank >=
\kappa, then this can be witnessed by indiscernible sequences of tuples
satisfying p. If p has dp-rank infinity, then this can be witnessed by
singletons (in any theory)
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