From the answer of Andreas Blass and comments of Ali Enayat on my question [Selective ultrafilter and bijective mapping](https://mathoverflow.net/questions/324254/selective-ultrafilter-and-bijective-mapping) it became clear that for any free ultrafilter $\mathcal{U}$ we have $\mathcal{U}\nsim\mathcal{U}\otimes\mathcal{U}$. Where $\mathcal{F}\otimes\mathcal{G}=\{X\subset A\times B~|~\{a\in A~|~ \{b\in B~|~ (a,b)\in X\} \in\mathcal{G}\}\in\mathcal{F}\}$ for any two filters $\mathcal{F}, \mathcal{G}$ on sets $A, B$ respectively. And $\mathcal{F}\sim\mathcal{G}$ means existing of bijection $f:A\to B$ such that $B\in\mathcal{F}\iff f(B)\in\mathcal{G}$. From [this link](https://dml.cz/bitstream/handle/10338.dmlcz/126182/MathBohem_122-1997-1_9.pdf) given by Ali Enayat I have also understood that $\mathcal{N}\nsim\mathcal{N}\otimes\mathcal{N}$ for the Freshet filter $\mathcal{N}$ but on any infinite set there exists a nontrivial idempotent filter $\mathcal{F}$, i.e. $\mathcal{F}\sim\mathcal{F}\otimes\mathcal{F}$. I couldn't get article referenced there as "On idempotent filters, Casopis Pest. Mat. 102 (1977), 412-418", in which the example of idempotent filter is given. So, I am asking the example of the idempotent filter or subsets family with finite intersections property. Can such a filter or family contain only infinite subsets?



The example of the idempotent filter or subsets family with finite intersections property
Following the link given by Martin Sleziak in comment I have got the answer:


There exists filter $\mathcal{F}$ with the following properties:


1. $\mathcal{N}\subset\mathcal{F}$
2. $\mathcal{F}\sim\mathcal{F}\otimes\mathcal{F}$


Existence proof is easy. But construction of $\mathcal{F}$ is complicated.


