Consider the class of all structures for a given signature in first-order logic. Let $S_i$ be a family of structures, and $\oplus S_i$ be the direct product of the family. You can extend the notion of a direct product to the cover the case of an empty family as follows. Let the underlying carrier set be a set with one element, $\lbrace x \rbrace$. For every function symbol, $f$, let $f(x, \ldots, x) = x$. For every relation symbol, $R$, let $R(x, \ldots, x)$ be true. If you consider the category of all structures with a given signature with homomorphisms as the morphisms, this is exactly the categorical direct product for an empty family.
Is it known what sentences are preserved by direct products, including the empty direct product? Most Horn sentences are still preserved. (The exception is Horn sentences with no positive literals.) The standard example of a non-Horn sentence preserved by direct products (the sentence "there exists an atom" in the signature of Boolean algebras) is not preserved by the empty direct product. It's tempting to conjecture that the sentences preserved by direct products, including the empty product, are exactly the sentences equivalent to a Horn sentence with one positive literal. Is there a known proof or counterexample?
