Isomorphic ultrafilters - open problems
Victor Porton
I have defined the relation "being isomorphic" for two filters and
proved that this relation is an equivalence relation as well as some
other simple theorems.
I published it at this blog entry:
http://portonmath.wordpress.com/2009/06/10/isomorphic-filters/
For me it remains an open problems: 1. whether any two non-trivial
ultrafilters are isomorphic; 2. whether any two non-trivial filters on
the set of natural numbers are isomorphic.
Maybe you know the answers for my open problems?
You may post comments for the above mentioned blog post.
Don't hesitate ask me questions about this.
hagman
If I got this right, given two non-trivial ultrafilters, you want
to find subsets in these filters such that the restriction of the
filters to these subsets are directly isomorphic (i.e. via a bijection
of these underlying sets).
Let N be the set of the naturals, R the set of the reals.
Let a_0 = { x c N | #(N\x) < oo } and refine this to an ultrafilter
a (using Axiom of Choice, of course).
Let b_0 = { x c R | #(R\x) <= #N } and refine this to an
ultrafilter b.
If a and b were directly isomorphic, then there should exist
sets A e a, B e b and a bijection f:A->B such that ...
But B is uncountable (else R\B e b_o c b) hence no such bijection
exists.
Hagen
Butch Malahide
> Hello mathematicians,
>
> I have defined the relation "being isomorphic" for two filters and
> proved that this relation is an equivalence relation as well as some
> other simple theorems.
>
> I published it at this blog entry:http://portonmath.wordpress.com/2009/06/10/isomorphic-filters/
>
> For me it remains an open problems: 1. whether any two non-trivial
> ultrafilters are isomorphic; 2. whether any two non-trivial filters on
> the set of natural numbers are isomorphic.
>
> Maybe you know the answers for my open problems?
No, there are many nonisomorphic nontrivial ultrafilters on the set N
of natural numbers.
There are 2^(2^{aleph_0}) ultrafilters on N. Each isomorphism class
contains at most 2^(aleph_0) ultrafilters. Therefore the number of
isomorphism classes is 2^(2^{aleph_0}). One of these isomorphism
classes consists of all the trivial (i.e. principal) ultrafilter on N.
Victor Porton
Why each isomorphism class contains at most 2^(aleph_0) ultrafilters?
I see only that the number of ultrafilters in a class is not more than
the number of bijection between subsets of N. What is this number?
Victor Porton
Oh, it seems that I've got it:
The number of bijections between any two subsets of N is no more that
(aleph_0)^(aleph_0) = 2^(aleph_0).
The number of all bijections between any two subsets of N is no more
than
2^(aleph_0) * 2^(aleph_0) = 2^(aleph_0).
Therefore Each isomorphism class contains at most 2^(aleph_0)
ultrafilters.
Q.E.D.