If two classes of structures are elementary, and one can be defined from the other, can the other be defined also?

Question

Suppose we have a certain class $C$ of binary relations $R$ which is an elementary class. Suppose also that we define a class $C'$ of binary relations $S$ in terms of $R$ by a first-order formula. For example, we could define that $S$ is the reflexive closure of $R$, or the symmetric closure of $R$. So that means we can form the definitional extension in the language of two binary relations $\{R,S\}$. So $C'$ is at least a pseudo-elementary class. It is obvious to me that if $R$ can be defined in terms of $S$, then the class $C'$ is in fact an elementary class. I am interested in the converse. That is, if $C'$ is an elementary class, can $R$ be defined in terms of $S$?

Answer 1

Let $C$ be the class of all binary relations (axiomatized by the empty theory), and let $C'$ be the class of empty relations (i.e., $S$ never holds). Then $S$ is definable from $R$ (by $\bot$) and $C'$ is elementary (axiomatized by $\forall x\forall y\,\lnot S(x,y)$), but we can't recover an arbitrary binary relation from the empty relation on the same set.

In fact, in the situation that you can define $R$ from $S$, the operation $(X,R)\mapsto (X,S)$ must be one-to-one. This happens pretty rarely - for example, the reflexive and symmetric closure operations are not one-to-one.