Let $\mathfrak{A}$ and $\Gamma$ be boolean lattices with $\Gamma$ being a boolean sublattice of $\mathfrak{A}$.
Let us denote $\mathfrak{F}(X)$ the set of filters on a poset $X$ ordered by set theoretic inclusion. Denote $\mathfrak{P}(X)$ the set of principal filters on a poset $X$ ordered by set theoretic inclusion.
Can we warrant that there exists an order embedding from $\mathfrak{F}(\Gamma)$ to $\mathfrak{F}(\mathfrak{A})$ which maps $\mathfrak{P}(\Gamma)$ into $\mathfrak{P}(\mathfrak{A})$ (with principal filters on $\Gamma$ mapped into corresponding (generated by the same element $x$ of $\Gamma$) principal filters on $\mathfrak{A}$)?
In the real example which I try to solve, $\mathfrak{A}$ is the set of binary relations between some (infinite) sets $A$ and $B$ and $\Gamma$ is the set of finite unions of binary cartesian products $X\times Y$ of subsets $X\in\mathscr{P}A$ and $Y\in\mathscr{P}B$ of these sets.
If this does not hold in general, does it hold for this special case?
The answer to your question is yes in general.
Let $S$ be a po (particially ordered) set and $C$ a po subset of $S$.
A filter for a po set is any down directed upper subset.
Define
$\operatorname{up}_S A = \{ x \in S \mid \exists a \in A: a \leq x \}$, $A$ subset $S$;
$\operatorname{up}_C A = \{ x \in C \mid \exists a \in C: a \leq x \}$, $A$ subset $C$;
$Fs = \{ F \mid F \text{ filter for }S \}$; $Fc = \{ F \mid F \text{ filter for }C \}$.
Theorem $\operatorname{up}S:Fc \rightarrow Fs$, $F \rightarrow \operatorname{up}_S F$ is subset-order embedding;
if $F$ principle filter in $Fc$, then $\operatorname{up}_S F$ is a principle filter in $Fs$.
Lemmas: for all $F$ in $Fc$, $\operatorname{up}_S F$ in $Fs$;
$F = C \cap \operatorname{up}_S F$.
Proof of theorem. Assume $F$, $G$ in $Fc$. Clearly if $F$ subset $G$, then $\operatorname{up}_S F$ subset $\operatorname{up}_S G$.
If $\operatorname{up}_S F$ subset $\operatorname{up}_S G$, then
$F = C \cap \operatorname{up}_S F$ subset $C \cap \operatorname{up}_S G$. Hence $\operatorname{up}_S$ is order embedding.
Let $F$ be a principle filter in $Fc$. Then
some $a$ in $A$ with $F = \operatorname{up}_C a$; $\operatorname{up}_S \operatorname{up}_C a = \operatorname{up}_S a$. Whence $\operatorname{up}_S F$ is a principle filter.