6
$\begingroup$

Let $T$ be an stable theory. Further we work in the monster model of $T^{eq}$. We say that a chain of types of the form $$tp(a_1/A_1)\subset tp(a_2/A_2) ... \subset tp(a_n/A_n)$$ is a forking chain if for every $1< i \le n$ the type $tp(a_i/A_i)$ forks over $A_{i-1}$.

What can we say about the length of forking chains?

For example in strongly minimal theories such a chain (in the home sort) has an maximum length of 1. Moreover, there exists no chain of length $|T|^+$ if and only if $T$ is simple. This is since forking has local character (every type does not fork over a set of size $<|T|$) if and only if $T$ is simple. For theory of Morley rank $N$ such a chain is bounded by $N$. But what about the lower bounds?

Can we find a chain of length $n$ in a theory with Morley rank $\ge n$?

Does any type $p$ of Morley rank $n$ start an forking chain of length $n$?

Do theories without Morley rank have chains of length $|T|$?

$\endgroup$
3
$\begingroup$

The answer is no to all three questions.

First notice that if there exists a maximal finite forking chain, then its length is the SU-rank (or U-rank in the stable context) of $tp(a_1/A_1)$. Since there are theories of Morley-rank $>1$ and U-rank $=1$, this gives negative answer to 1 and 2.

Then note that in a supersimple theory there is no infinite forking chain. As $\bigcup_i tp(a_i/A_i)$ would not fork over some finite $B$ and this would be contained in $A_N$ for $N$ sufficiently big. Hence strictly superstable theories are a counter-example to question 3.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.