I have a finite set $X$ and a finite family of subsets $X_i \subset X$, $0 <= i < n$, $n \in \mathbb{N}$.
What can we say about the size of the transitive hull of this family with regards to (binary) intersection? Can it be bounded somehow given any constraint information about the initial family?
To get more concrete, I think about creating a join semilattice where $\#X$ is roughly on the order of $n$, and small, like $n < 50$. I want to answer queries against a relational database, posed in the form of conjunctions of first-order logic predicates. Creating this join semilattice of course requires to keep the number of elements under control.
The size is bounded from below by $1$ and from above by $2^n$.
The lower bound is obvious and attained for the family $X_i = \emptyset$. To see that the upper bound is an actual bound, observe that the set of intersections of the $X_i$ can be written as $$\left\{ \bigcap \limits_{i \in S} X_i \mid S \subset \{0, \ldots, n - 1\}\right\}.$$
Now note that there are $2^n$ subsets of $\{0, \ldots, n - 1\}$.
To see that this bound is actually obtained, let $X_i$ be the set of all numbers $x$ with $0 \le x < 2^n$ that have a $0$ on the $i$-th position in their binary representation. Then for any two sets $S \ne S'$ we have $\bigcap \limits_{i \in S} X_i \ne \bigcap \limits_{i \in S'} X_i$.