434 research outputs found
On Quasiminimal Excellent Classes
A careful exposition of Zilber's quasiminimal excellent classes and their
categoricity is given, leading to two new results: the L_w1,w(Q)-definability
assumption may be dropped, and each class is determined by its model of
dimension aleph_0.Comment: 16 pages. v3: correction to the statement of corollary 5.
On properties of (weakly) small groups
A group is small if it has countably many complete -types over the empty
set for each natural number n. More generally, a group is weakly small if
it has countably many complete 1-types over every finite subset of G. We show
here that in a weakly small group, subgroups which are definable with
parameters lying in a finitely generated algebraic closure satisfy the
descending chain conditions for their traces in any finitely generated
algebraic closure. An infinite weakly small group has an infinite abelian
subgroup, which may not be definable. A small nilpotent group is the central
product of a definable divisible group with a definable one of bounded
exponent. In a group with simple theory, any set of pairwise commuting elements
is contained in a definable finite-by-abelian subgroup. First corollary : a
weakly small group with simple theory has an infinite definable
finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal
solvable group A of derived length n is contained in an A-definable almost
solvable group of class n
Invariant measures concentrated on countable structures
Let L be a countable language. We say that a countable infinite L-structure M
admits an invariant measure when there is a probability measure on the space of
L-structures with the same underlying set as M that is invariant under
permutations of that set, and that assigns measure one to the isomorphism class
of M. We show that M admits an invariant measure if and only if it has trivial
definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary
finite tuple of M fixes no additional points. When M is a Fraisse limit in a
relational language, this amounts to requiring that the age of M have strong
amalgamation. Our results give rise to new instances of structures that admit
invariant measures and structures that do not.Comment: 46 pages, 2 figures. Small changes following referee suggestion
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