This question isn't category theory; but, category theoreticians tend to be interested in mathematical structure, so I thought the answer might exist within that knowledge base.
Given two mathematical structures $X$ and $Y$ with the same pattern of airities, there is a natural notion of homomorphism $X \rightarrow Y$, described here, called a structural homomorphism. However, this notion lacks "robustness."
For instance, suppose that $X$ and $Y$ are partially ordered sets and that $f$ is a structural homomorphism $f : X \rightarrow Y$. Then $f$ is an order homomorphism. Suppose also that $X$ and $Y$ happen to be lattices. Then we can add meets and joins to the data of $X$ and $Y$, thereby obtaining new structures $X'$ and $Y'$. However, its feasible that $f$ might fail to be a homomorphism $X' \rightarrow Y'$, since not every order homomorphism is a lattice homomorphism.
So in general, if $f$ is a structural homomorphism $X \rightarrow Y$, and we extend $X$ and $Y$ by defining new relations/operations in terms of the old ones, thereby obtaining new structures $X'$ and $Y'$, well even if the new relations/operations are defined in terms of the old ones, using exactly the same definitions, nonetheless $f$ may fail to be a homomorphism $X' \rightarrow Y'$.
Now I originally thought that the notion of a "structural embedding" described here in that same article, might be robust with respect to the defining of new relations/operations. But that's completely wrong.
So my question is, what's more robust than a structural homomorphism?
Note that the example given by Zev in the previous thread is a structural embedding. So the same example shows that embedding between posets need not preserve meet/join.
For a stronger notion you should probably talk about theories as well, and require the embedding to preserve truth values of sentences, in the sense of an elementary embedding. Then you can extend definable operations from one structure to the next.