It is well-known that Horn sentences are "preserved" under products (see for instance Show that the direct product of structures satisfies a Horn sentence).
I was wondering what happens with the converse. I.e. if a Horn sentence is true in the product is it true in the components? What if all components are equal (i.e. we have a power structure)?
No. For example, "There are at least four elements" is expressible as a Horn sentence but does not "descend to factors" (even in power structures) for obvious reasons.
To see that "There are at least four elements" is indeed expressible as a Horn sentence, we proceed as follows:
For each pair of distinct variables $i,j\in\{1,2,3,4\}$, the expression $x_i\not=x_j$ is a negated atomic formula and so is a Horn formula. (Clause $2$ with $n=1$ in the linked post)
The arbitrary conjunction of Horn formulas is again a Horn formula, so $$\bigwedge_{1\le i<j\le 4}x_i\not=x_j$$ is a Horn formula. (Clause $3$, repeatedly)
Finally, we can existentially generalize, so that $$\exists x_1,x_2,x_3,x_4\left(\bigwedge_{1\le i<j\le 4}x_i\not=x_j\right)$$ is a Horn sentence as claimed. (Clause 5, repeatedly)
EDIT: Note that while the above shows that it is easy to express "There are at least $k$ elements" for each $k$ as a Horn sentence, things are murkier if we want to say "There are not exactly $k$ elements" - while the preservation theorem says that this can't be done for composite $k$, it turns out to be doable-but-tricky for prime $k$. See this MO thread.
