Suppose that $P$ and $Q$ are two posets and $f:P\to Q$ is a homomorphism (a.k.a., $f(x)\le f(y)$ whenever $x\le y$). Given a chain $C\subset P$, the image $f(C)$ automatically is a chain as well. However, if $C$ is maximal (i.e., it is not properly contained in some other chain) then the image $f(C)$ need not be maximal, even if $f$ is onto.
I am interested in knowing if there are any natural conditions on $P$, $Q$, and $f$ which guarantee that maximal chains in $P$ get mapped to chains which continue to be maximal in $Q$.
For the application I have in mind, $P$, and $Q$ are actually finite lattices and $f$ is a (surjective) lattice homomorphism. Though, I'm hoping I can get away with weaker assumptions.