# A new kind of product of funcoids

The following is one of a few (possibly non-equivalent) definitions of products of funcoids:

Definition Let $f$ be an indexed family of funcoids. Let $\mathcal{F}$ be a filter on $\mathrm{dom}\, f$. $a \mathrel{\left[ \prod^{[\mathcal{F}]} f \right]} b \Leftrightarrow \exists N \in \mathcal{F} \forall i \in N : \mathrm{Pr}^{\mathsf{RLD}}_i\, a \mathrel{[f_i]} \mathrm{Pr}^{\mathsf{RLD}}_i\, b.$
for atomic reloids $a$ and $b$.

Today I have proved that this really defines a funcoid. Currently the proof is present in draft of the second volume of my book,

A probably especially interesting case is if $\mathcal{F}$ is the cofinite filter. In this way we get something similar to Tychonoff product of topological spaces.

This may possibly have some use in study of compact funcoids.