Motivation (skippable content).$\newcommand{\M}{\mathfrak{M}}$ $\newcommand{\L}{\mathscr{L}}$ I had difficulty to understand the definition in my lecture note of "structure definable to another one". Luckily, the definition of Wikipedia (they use interpretable for definable) is very clear to me, however their definition is dual to the one of my lecture note (namely surjective $\to$ injective, preimage $\to$ image).
I tried to reformulate the definition in my lecture note in a dual way of the Wikipedia's one as follows:
Let $\M$ be an $\L$-structure and $\M'$ be an $\L'$-structure. If there is an injection $\varphi : \M \hookrightarrow \M'$ such that $\varphi(M) \subseteq (M')^n$ (for some $n$) such that the $f^k$-image $f(X)$ of every subset $X \subseteq M^k$ definable without parameter is definable with parameters in $M'$. Then we say that $\M$ is definable in $\M'$ with parameters.
I also adapted the sufficient condition to prove that image preserve definability of definable sets of $\M$, namely:
To show that the image of every definable-without-parameter set in $M$ is definable in $M'$ with parameters, it suffices to check that the images of the following definable sets:
- the domain of $M$;
- the diagonal of $M$;
- every $\{c^\M\}$ for every constant $c$;
- the graph of every function $f^\M$;
- every relation $P^\M$.
But I realized that this condition may not be sufficient with my definition. Indeed, with Wikipedia's definition, as preimage preserves union, intersection and complement it suffices to prove that the preimage preserves the definable sets above to show that every definable sets are preserved by preimage (that's because the set of definable subsets is the smallest set containing above sets and closed under the Boolean operation aforementioned).
However, with my definition, even though an injective function's image preserves union and intersection, it doesn't preserve complement (it has to be also surjective). So I cannot ensure that, given a set $A$ definable in $\M$ such that $\varphi(A)$ is definable in $\M'$, $\varphi(A^C)$ is definable in $\M'$.
Question: Is my definition correct? If not, is it possible to adapt it as my attempt?
EDIT: As suggested @tomasz in comment, I confused definability and interpretability. So my question turned to be
Question: What's the correct definition of definability ?
EDIT 2:
This is my new phrasing of the definition, please tell me if it is correct: $\newcommand{\M}{\mathfrak{M}}$ $\newcommand{\N}{\mathfrak{N}}$ $\newcommand{\Def}{\mathsf{Def}}$
Definition (Definability and interdefinability). Let $\M$ be an $\L_1$-structure and $\N$ an $\L_2$-structure. If there is an injection $\varphi: \Def(\M, M) \hookrightarrow \Def(\N, N)$ such that
- $\varphi(M) \subseteq N^k$, for some $k$, can be endowed with an $\L_1$-structure $\M'$ isomorphic to $\M$;
- symbols interpreted in $\M'$ (i.e. $\{c^{\M'}\}, R^{\M'}$ and graph of $f^{\M'}$) are definable in $\N$.
Then we say that $\M$ is definable in $\N$ with parameters. If $\M$ and $\N$ are definable in each other, with say that they are interdefinable.
($\Def(\M, M)$ is the set of subsets $M$-definable in $\M$).