How many sets of vectors can be represented as the solutions of a Horn-SAT instance?
$\def\pw{\mathcal P}$
It is an easy exercise to show that if we identify Boolean assignments to $n$ variables with subsets of $[n]=\{0,\dots,n-1\}$, then the solution sets of Horn CNFs are exactly the families $S\subseteq\pw([n])$ that are closed under nonempty intersections.

The picture is a little nicer for conjunctions of clauses each of which contains exactly one positive literal. Their solution sets are families closed under arbitrary intersections (where in particular the empty intersection is the full set), also known as closure systems or Moore families. These are in 1–1 correspondence with closure operators (functions $C\colon\pw([n])\to\pw([n])$ satisfying $X\subseteq C(Y)\iff C(X)\subseteq C(Y)$), consequence relations (relations ${\vdash}\subseteq\pw([n])\times[n]$ satisfying a handful of natural conditions), and I guess there are more similar combinatorial characterizations (https://oeis.org/A102896 or https://oeis.org/A102897 mention semilattices).

These two numbers only differ by a factor of two (given a closure system $S$, we may either retain $[n]\in S$ or throw it out without affecting anything else).

Alekseev proved that the number $\alpha(n)$ of closure systems on an $n$-element set satisfies

$$2^{\binom{n}{\lfloor n/2\rfloor}}\le\alpha(n)\le2^{\binom{n}{\lfloor n/2\rfloor}(1+o(1))}.$$

See http://www.renyi.hu/~ohkatona/paper_72.pdf for related information.