# The example of the idempotent filter or subsets family with finite intersections property

From the answer of Andreas Blass and comments of Ali Enayat on my question 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 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 Katětov's paper you mention can be found at eudml or at dml.cz. (Both locations are easy to find using Google Scholar.) – Martin Sleziak Mar 7 '19 at 8:31
• Thank you. I have got the answer. – ar.grig Mar 8 '19 at 14:20

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.