I have a conjecture relating statements about inclusions of sets to corresponding statements about inclusions of linear subspaces.
More specifically, consider a formula
\begin{equation*}
\phi \equiv t_1 \subseteq t_2,
\end{equation*}
where $t_1$ and $t_2$ are terms of sets, formed recursively from set variables, binary intersection/union and intersection/union of families of sets (set variables can be indexed). For example,
\begin{equation*}
A \cap B \subseteq A
\end{equation*}
\begin{equation*}
A \subseteq A \cup B
\end{equation*}
\begin{equation*}
\bigcap_{i \in I}(A_i \cap B_i) \subseteq \bigcap_{i \in I} A_i \cup \bigcup_{i \in I} B_i
\end{equation*}
are such formulae.
Replacing $\cup$ by $+$ and $\bigcup$ by $\sum$, $\phi$ may be interpreted as a statement $\phi'$ about inclusions of linear subspaces of a single vector space $V$. I conjecture that
$\phi$ is a tautology in the universe of sets (i.e. the formula is true for every family of sets assigned to each variable) iff $\phi'$ is a tautology in the universe of subspaces of $V$ for each vector space $V$.
I'm not sure how to prove this tough (is it even correct?). Is this obvious from some general abstract nonsense, or is syntactic induction the right tool in this situtation?
$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ holds for subsets of a set, but the corresponding equation $U \cap (V + W) = (U \cap V) + (U \cap W)$ does not hold for subspaces of a vector space. Another difference is that every subset of a set has a unique complement, but subspaces of a vector space have lots of complements.