51 research outputs found
Elimination of Hyperimaginaries and Stable Independence in simple CM-trivial theories
International audienceIn a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial theory, the independence relation is stable
Automorphism groups over a hyperimaginary
In this paper we study the Lascar group over a hyperimaginary e. We verify
that various results about the group over a real set still hold when the set is
replaced by e. First of all, there is no written proof in the available
literature that the group over e is a topological group. We present an
expository style proof of the fact, which even simplifies existing proofs for
the real case. We further extend a result that the orbit equivalence relation
under a closed subgroup of the Lascar group is type-definable. On the one hand,
we correct errors appeared in the book, "Simplicity Theory" [6, 5.1.14-15] and
produce a counterexample. On the other, we extend Newelski's Theorem in "The
diameter of a Lascar strong type" [12] that `a G-compact theory over a set has
a uniform bound for the Lascar distances' to the hyperimaginary context.
Lastly, we supply a partial positive answer to a question raised in "The
relativized Lascar groups, type-amalgamations, and algebraicity" [4, 2.11],
which is even a new result in the real context
SHELAH-STRONG TYPE AND ALGEBRAIC CLOSURE OVER A HYPERIMAGINARY (Model theoretic aspects of the notion of independence and dimension)
We characterize Shelah-strong type over a hyperimagianary with the algebraic closure of a hyperimaginary. Also, we present and take a careful look at an example that witnesses acl[eq](ℯ) is not interdefinable with acl(ℯ) where ℯ is a hyperimaginary
Plus ultra
International audienceWe define a reasonably well-behaved class of ultraimaginaries, i.e.\ classes modulo invariant equivalence relations, called {\em tame}, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If is an ultraimaginary definable over a tuple with , then is eliminable up to rank . Finally, we prove some uniform versions of the weak canonical base property
- …