Several ways to apply a (multivalued) multiargument function to a family of filters

Problem's Description

As described in my book Algebraic General Topology. Book 1, to every relation $f$ it corresponds a multifuncoid (to a relation it corresponds a staroid and to the staroid it corresponds a multifuncoid).

Let $\mathcal{X}$ be an indexed family of filters on sets. Which of the below items are always pairwise equal?

1. The above-mentioned multifuncoid applied to $\mathcal{X}$

2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters $\mathcal{X}$.

3. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters $\mathcal{X}$.

4. $\bigcap_{F\in\operatorname{up}^{\mathrm{FCD}}\prod^{\mathrm{Strd}}\mathcal{X}}\langle f \rangle F$.

1. Book Algebraic General Topology. Volume 1

   • Victor Lvovich Porton

   year of publication: 2018fulltext