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

Definition Let $latex f$ be an indexed family of funcoids. Let $latex \mathcal{F}$ be a filter on $latex \mathrm{dom}\, f$. $latex 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 $latex a$ and $latex b$.

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 $latex \mathcal{F}$ 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.

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