I have re-defined *filter rebase*. Now it is defined for arbitrary filter $latex \mathcal{A}$ on some set $latex \mathrm{Base}(\mathcal{A})$ and arbitrary set $latex A$.

The new definition is: $latex \mathcal{A}\div A = \{ X\in\mathscr{P}A \mid \exists Y\in\mathcal{A}: Y\cap A\subseteq X \}$.

It is shown that for the special case of $latex \forall X\in\mathcal{A}:X\subseteq A$ the new definition is equal to the old definition that is $latex \mathcal{A}\div A = \{ X\in\mathscr{P}A \mid \exists Y\in\mathcal{A}: Y\subseteq X \}$.

See my book (updated), chapter “Orderings of filters in terms of reloids”, for details.

The new definition is useful for studying restrictions and embeddings of funcoids and reloids.