62 research outputs found
Groups definable in two orthogonal sorts
This work can be thought of as a contribution to the model theory of group extensions. We study the groups G which are interpretable in the disjoint union of two structures (seen as a two-sorted structure). We show that if one of the two structures is superstable of finite Lascar rank and the Lascar rank is definable, then G is an extension of a group internal to the (possibly) unstable sort by a definable normal subgroup internal to the stable sort. In the final part of the paper we show that if the unstable sort is an o-minimal expansion of the reals, then G has a natural Lie structure and the extension is a topological cover
Pseudofinite structures and simplicity
We explore a notion of pseudofinite dimension, introduced by Hrushovski and
Wagner, on an infinite ultraproduct of finite structures. Certain conditions on
pseudofinite dimension are identified that guarantee simplicity or
supersimplicity of the underlying theory, and that a drop in pseudofinite
dimension is equivalent to forking. Under a suitable assumption, a
measure-theoretic condition is shown to be equivalent to local stability. Many
examples are explored, including vector spaces over finite fields viewed as
2-sorted finite structures, and homocyclic groups. Connections are made to
products of sets in finite groups, in particular to word maps, and a
generalization of Tao's algebraic regularity lemma is noted
Dimensional groups and fields
We shall define a general notion of dimension, and study groups and rings
whose interpretable sets carry such a dimensio. In particular, we deduce chain
conditions for groups, definability results for fields and domains, and show
that pseudofinite groups contain big finite-by-abelian subgroups, and
pseudofinite groups of dimension 2 contain big soluble subgroups
On Variants of CM-triviality
We introduce a generalization of CM-triviality relative to a fixed invariant
collection of partial types, in analogy to the Canonical Base Property defined
by Pillay, Ziegler and Chatzidakis which generalizes one-basedness. We show
that, under this condition, a stable field is internal to the family, and a
group of finite Lascar rank has a normal nilpotent subgroup such that the
quotient is almost internal to the family
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