14 research outputs found
The stability of finite sets in dyadic groups
We show that there is an absolute such that any subset of
of size is -stable in the sense of Terry
and Wolf. By contrast a size arithmetic progression in the integers is not
-stable.Comment: 9 pp; corrected some errors and expanded the introductio
A preservation theorem for theories without the tree property of the first kind
We prove that the NTP property of a geometric theory is inherited by
theories of lovely pairs and -structures associated to . We also provide
a class of examples of nonsimple geometric NTP theories
Natural models of theories of green points
We explicitly present expansions of the complex field which are models of the
theories of green points in the multiplicative group case and in the case of an
elliptic curve without complex multiplication defined over . In
fact, in both cases we give families of structures depending on parameters and
prove that they are all models of the theories, provided certain instances of
Schanuel's conjecture or an analogous conjecture for the exponential map of the
elliptic curve hold. In the multiplicative group case, however, the results are
unconditional for generic choices of the parameters
Generic expansions and the group configuration theorem
We exhibit a connection between geometric stability theory and the
classification of unstable structures at the level of simplicity and the
- gap. Particularly, we introduce generic
expansions of a theory associated with a definable relation of
, which can consist of adding a new unary predicate or a new equivalence
relation. When is weakly minimal and is a ternary fiber algebraic
relation, we show that is a well-defined theory,
and use one of the main results of geometric stability theory, the
\textit{group configuration theorem} of Hrushovski, to give an exact
correspondence between the geometry of and the classification-theoretic
complexity of . Namely, is , and
exactly when is geometrically equivalent to the graph of
a type-definable group operation; otherwise, is either simple (in the
predicate version of ) or (in the equivalence
relation version.) This gives us new examples of strictly
theories.Comment: 25 pages; 1 figur
Embedded Finite Models beyond Restricted Quantifier Collapse
We revisit evaluation of logical formulas that allow both uninterpreted
relations, constrained to be finite, as well as interpreted vocabulary over an
infinite domain: denoted in the past as embedded finite model theory. We extend
the analysis of "collapse results": the ability to eliminate first-order
quantifiers over the infinite domain in favor of quantification over the finite
structure. We investigate several weakenings of collapse, one allowing
higher-order quantification over the finite structure, another allowing
expansion of the theory. We also provide results comparing collapse for unary
signatures with general signatures, and new analyses of collapse for natural
decidable theories