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-1 of filter $a$ into filter $b$ a function $f$ such that $f \left[ a \right] \subseteq b$ and super-embedding-2 of filter $a$ into filter $b$ a function $f$ such that $f \left[ a \right] = b$.

Let define preorders $\leqslant_1$ and $\leqslant_2$ on the set of filters:

$b \leqslant_1 a$ if there are super-embedding-1 from $a$ to $b$ and $b \leqslant_2 a$ if there are super-embedding-2 from $a$ to $b$.

Question 1: $\leqslant_1$ is the same as $\leqslant_2$?

Filters $a$ and $b$ are isomorphic if exists a bijective super-embedding-2 $f$ from $a$ to $b$ such that $f^{- 1}$ is super-embedding-2 from $b$ to $a$. For two other equivalent characterizations of isomorphic filters see this blog post and this blog post (the second blog post requires this article).

Being isomorphic is an equivalence relation. I will call classes of filters equivalence classes under the being isomorphic relation. I will call classes of ultrafilters these classes of filters which contain ultrafilters.

Further I will denote $i = 1, 2$. So every open problem below is in fact two problems.

Question 2: Is $\leqslant_i$ for ultrafilters the same as Rudin-Keisler order (paragraph 9 of Comfort and Negrepontis ``The Theory of Ultrafilters'') of ultrafilters? If not, how they are related?

Question 3: Is the preorder of classes of filters induced by $\leqslant_i$ a partial order?

Question 4: Is the preorder of classes of ultrafilters induced by $\leqslant_i$ a partial order?

Question 5: If it is a partial order, is it a linear order?

Question 6: If it is a linear order, is it a well-order (or maybe anti-well-order)?

Question 7: If in the above definition of isomorphic filters super-embedding-2 is replaced with super-embedding-1, does it remain equivalent to the above definition?