46 research outputs found
A note on Lascar strong types in simple theories
Let T be a countable, small simple theory. In this paper, we prove for such
T, the notion of Lascar Strong type coincides with the notion of a strong
type,over an arbitrary set
The Lascar groups and the 1st homology groups in model theory
Let be a strong type of an algebraically closed tuple over
B=\acl^{\eq}(B) in any theory . Depending on a ternary relation \indo^*
satisfying some basic axioms (there is at least one such, namely the trivial
independence in ), the first homology group can be introduced,
similarly to \cite{GKK1}. We show that there is a canonical surjective
homomorphism from the Lascar group over to . We also notice that
the map factors naturally via a surjection from the `relativised' Lascar group
of the type (which we define in analogy with the Lascar group of the theory)
onto the homology group, and we give an explicit description of its kernel. Due
to this characterization, it follows that the first homology group of is
independent from the choice of \indo^*, and can be written simply as
. As consequences, in any , we show that
unless is trivial, and we give a criterion for the equality of stp and
Lstp of algebraically closed tuples using the notions of the first homology
group and a relativised Lascar group. We also argue how any abelian connected
compact group can appear as the first homology group of the type of a model.Comment: 30 pages, no figures, this merged with the article arXiv:1504.0772
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
Transitivity, lowness, and ranks in NSOP theories
We develop the theory of Kim-independence in the context of NSOP
theories satsifying the existence axiom. We show that, in such theories,
Kim-independence is transitive and that \ind^{K}-Morley sequences witness
Kim-dividing. As applications, we show that, under the assumption of existence,
in a low NSOP theory, Shelah strong types and Lascar strong types
coincide and, additionally, we introduce a notion of rank for NSOP
theories