35 research outputs found
Categoricity in Quasiminimal Pregeometry Classes
Quasiminimal pregeometry classes were introduces by Zilber [2005a] to isolate
the model theoretical core of several interesting examples. He proves that a
quasiminimal pregeometry class satisfying an additional axiom, called
excellence, is categorical in all uncountable cardinalities. Recently Bays et
al. [2014] showed that excellence follows from the rest of axioms. In this
paper we present a direct proof of the categoricity result without using
excellence
The algebraic numbers definable in various exponential fields
We prove the following theorems: Theorem 1: For any E-field with cyclic
kernel, in particular or the Zilber fields, all real abelian
algebraic numbers are pointwise definable. Theorem 2: For the Zilber fields,
the only pointwise definable algebraic numbers are the real abelian numbers
Finding groups in Zariski-like structures
We study quasiminimal classes, i.e. abstract elementary classes (AECs) that arise from a quasiminimal pregeometry structure. For these classes, we develop an independence notion, and in particular, a theory of independence in M^{eq}. We then generalize Hrushovski's Group Configuration Theorem to our setting. In an attempt to generalize Zariski geometries to the context of quasiminimal classes, we give the axiomatization for Zariski-like structures, and as an application of our group configuration theorem, show that groups can be found in them assuming that the pregeometry obtained from the bounded closure operator is non-trivial. Finally, we study the cover of the multiplicative group of an algebraically closed field and show that it provides an example of a Zariski-like structure