I'm a bit confused about the use of " X preserves Y " in model theory, where X is a placeholder for a map of L-structures of some kind (e.g. homomorphism/ embedding/ isomorphism) and Y is a placeholder for some subset of terms or formulae (e.g. positive formulae/ atomic formulae) I have seen two distinct usages
A map $\phi: A \to B$ preserves Y iff $\forall \psi \in Y \forall \bar a \ A \models \phi(\psi(\bar a)) \iff B \models \psi(\phi(\bar a))$ and
A map $\phi: A \to B$ preserves Y iff $\forall \psi \in Y \forall \bar a \ A \models \phi(\psi(\bar a)) \implies B \models \psi(\phi(\bar a))$.
Here is an example that I think makes the issue I have clear:
According to Wikipedia, $\phi$ is a homomorphism iff $\phi$ preserves constants and functions ( so far this is clear to me) and $\forall \bar a \forall R\in \operatorname{Pred}(L) \ \bar a \in R^A \implies \phi(\bar a) \in R^B$. In this case would one say $\phi$ preserves relations or not?
The first has an "iff" the latter has "if... then...". Does anyone know what the convention around this terminology is? Many thanks!
Strictly speaking the latter (narrower) version is correct, and the term "reflects" is used for the dual notion. So, for example, surjective homomorphisms preserve but do not necessarily reflect universally-quantified equations, and general homomorphisms preserve but do not necessarily reflect relations.
That said, the conflation of "preserves" and "preserves and reflects" is fairly common (unfortunately, I would argue).