316,970 research outputs found
Simple structures axiomatized by almost sure theories
In this article we give a classification of the binary, simple,
-categorical structures with SU-rank 1 and trivial pregeometry. This is
done both by showing that they satisfy certain extension properties, but also
by noting that they may be approximated by the almost sure theory of some sets
of finite structures equipped with a probability measure. This study give
results about general almost sure theories, but also considers certain
attributes which, if they are almost surely true, generate almost sure theories
with very specific properties such as -stability or strong minimality.Comment: 27 page
Some Definability Results in Abstract Kummer Theory
Let be a semiabelian variety over an algebraically closed field, and let
be an irreducible subvariety not contained in a coset of a proper algebraic
subgroup of . We show that the number of irreducible components of
is bounded uniformly in , and moreover that the bound is
uniform in families .
We prove this by purely Galois-theoretic methods. This proof applies in the
more general context of divisible abelian groups of finite Morley rank. In this
latter context, we deduce a definability result under the assumption of the
Definable Multiplicity Property (DMP). We give sufficient conditions for finite
Morley rank groups to have the DMP, and hence give examples where our
definability result holds.Comment: 21 pages; minor notational fixe
On the canonical base property
We give an example of a finite rank, in fact aleph-1 categorical theory where
the CBP (canonical base property) does not hold. We include a "group-like"
example. We also prove, in a finite Morley rank context, if all definable
Galois groups are "rigid" then the theory has the CBP (in a strong sense).Comment: 15 page
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