Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

$G$ is a group with a simple first order theory $T$ as defined by Shelah, hence equiped with a "nice" notion of independence. $G$ also has generic elements. I write $acl^n(A)$ for the set of elements in $M$ lying in a finite $A$-definable set of size at most $n$, and $acl(A)$ for the union of $acl^n(A)$ when $n$ ranges over all natural numbers. Let $a$ and $b$ be independent elements over $0$. One has

$$acl(a)\cap acl(b)=acl(\emptyset)$$

In particular, this holds if $a$ and $b$ are independent generic elements of $G$.

Question 1 : If $a$ and $b$ are generic, does the following equality hold? $$acl^n(a)\cap acl^n(b)=acl^n(\emptyset)$$

As I expect the answer to question 1 to be no, let me ask :

Question 2 : Is there a constant $k$ (depending only on $T$ and $n$) such that for all independent generics $a$ and $b$ over $0$, one would have $$acl^n(a)\cap acl^n(b)\subset acl^{k.n}(\emptyset)$$

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.