Let $U$ be a set. I denote $\mathfrak{A}$ the lattice of filters on $U$ ordered reverse to set theoretic inclusion of filters. I denote $\bigvee$ and $\bigwedge$ correspondingly the supremum and infimum on $\mathfrak{A}$.

Let $N$ be a (possibly infinite) index set. Let $f$ be an $N$-ary relation.

I denote $\langle f\rangle_k L = \{ x \mid L\cup\{(k,x)\}\in f \}$ for $k\in N$.

I denote $\langle \upuparrows f\rangle_k \mathcal{L} = \bigwedge_{L\in\prod \mathcal{L}} \langle f\rangle L$ for every $(N\setminus\{k\})$-indexed family $\mathcal{L}$ of filters on $U$.

Conjecture $$\langle \upuparrows f\rangle_k \mathcal{L} = \bigvee_{a\in \prod_{i\in N\setminus\{k\}} \operatorname{atoms} \mathcal{L}_i} \langle \upuparrows f\rangle_k a,$$

where $\operatorname{atoms} \mathcal{X}$ is the set of ultrafilters over $\mathcal{X}$.

Let $U$ be a set. I denote $\mathfrak{A}$ the lattice of filters on $U$ ordered reverse to set theoretic inclusion of filters. I denote $\bigvee$ and $\bigwedge$ correspondingly the supremum and infimum on $\mathfrak{A}$.

Let $N$ be a (possibly infinite) index set. Let $f$ be an $N$-ary relation.

I denote $\langle f\rangle_k L = \{ x \mid L\cup\{(k,x)\}\in f \}$ for $k\in N$.

I denote $\langle \upuparrows f\rangle_k \mathcal{L} = \bigwedge_{L\in\prod \mathcal{L}} \uparrow \langle f\rangle_k L$ for every $(N\setminus\{k\})$-indexed family $\mathcal{L}$ of filters on $U$ (here $\uparrow X$ is the principal filter corresponding to $X$).

Conjecture $$\langle \upuparrows f\rangle_k \mathcal{L} = \bigvee_{a\in \prod_{i\in N\setminus\{k\}} \operatorname{atoms} \mathcal{L}_i} \langle \upuparrows f\rangle_k a,$$

where $\operatorname{atoms} \mathcal{X}$ is the set of ultrafilters over $\mathcal{X}$.