Let $U$ is a set. I will speak about filters on this set.

If $f$ is a function and $a$ is a filter then I define $f \left[ a \right]$ as the filter whose base is $\lbrace f[A] | A \in a \rbrace$.

I will call super-embedding of filter $a$ into filter $b$ a function $f$ such that that $f[a] = b$.

$b \leqslant a$ if there are super-embedding from $a$ to $b$.

I will call filters $a$ and $b$ directly isomorphic when there are a bijective super-embedding from $a$ to $b$.

I will call filters $a$ and $b$ isomorphic iff there exist sets $A$ and $B$ such that $a\cap\mathcal{P}A$ is directly isomorphic to $b\cap\mathcal{P}B$.

Question If $a\le b$ and $b\le a$ then $a$ and $b$ are isomorphic, for every filters $a$ and $b$?

You may consult my article "Orderings of filters in terms of reloids" (at this Web page) about what I already know about ordering and isomorphism of filters.

Let $U$ is a set. I will speak about filters on this set.

If $f$ is a function and $a$ is a filter then I define $f \left[ a \right]$ as the filter whose base is $\lbrace f[A] | A \in a \rbrace$.

I will call super-embedding of filter $a$ into filter $b$ a function $f$ such that that $f[a] = b$.

$b \leqslant a$ if there are super-embedding from $a$ to $b$.

I will call filters $a$ and $b$ directly isomorphic when there are a bijective super-embedding from $a$ to $b$.

I will call filters $a$ and $b$ isomorphic iff there exist sets $A\in a$ and $B\in b$ such that $a\cap\mathcal{P}A$ is directly isomorphic to $b\cap\mathcal{P}B$.

Question If $a\le b$ and $b\le a$ then $a$ and $b$ are isomorphic, for every filters $a$ and $b$?

You may consult my article "Orderings of filters in terms of reloids" (at this Web page) about what I already know about ordering and isomorphism of filters.