44 research outputs found
Conant-independence and generalized free amalgamation
We initiate the study of a generalization of Kim-independence,
Conant-independence, based on the "strong Kim-dividing" of Kaplan, Ramsey and
Shelah. We introduce an axiom on stationary independence relations essentially
generalizing the "freedom" axiom in some of the free amalgamation theories of
Conant, and show that this axiom provides the correct setting for carrying out
arguments of Chernikov, Kaplan and Ramsey on theories
relative to a stationary independence relation. Generalizing Conant's results
on free amalgamation to the limits of our knowledge of the
hierarchy, we show using methods from Conant as well as our previous work that
any theory where the equivalent conditions of this local variant of
holds is either or
and is either simple or , and observe that these theories give
an interesting class of examples of theories where Conant-independence is
symmetric, including all of Conant's examples, the small cycle-free random
graphs of Shelah and the (finite-language) -categorical Hrushovski
constructions of Evans and Wong.
We then answer a question of Conant, showing that the generic functional
structures of Conant and Kruckman are examples of non-modular free amalgamation
theories, and show that any free amalgamation theory is or
, while an free amalgamation theory is
simple if and only if it is modular.
Finally, we show that every theory where Conant-independence is symmetric is
. Therefore, symmetry for Conant-independence gives the next
known neostability-theoretic dividing line on the hierarchy
beyond . We explain the connection to some established open
questions.Comment: 23 page
A New Kim's Lemma
Kim's Lemma is a key ingredient in the theory of forking independence in
simple theories. It asserts that if a formula divides, then it divides along
every Morley sequence in type of the parameters. Variants of Kim's Lemma have
formed the core of the theories of independence in two orthogonal
generalizations of simplicity - namely, the classes of NTP2 and NSOP1 theories.
We introduce a new variant of Kim's Lemma that simultaneously generalizes the
NTP2 and NSOP1 variants. We explore examples and non-examples in which this
lemma holds, discuss implications with syntactic properties of theories, and
ask several questions