I was reading about isomorphisms and homomorphisms on general structures, and first came across the definition of an injective homomorphism, or an embedding. This made me curious, is it possible for two structures $A$ and $B$ to be embeddable in each other, yet no isomorphism exists between them?
After some looking around, I let the structures be $A=\mathbb{R}$ and $B=[-1,1]$ with $f\colon [-1,1]\to\mathbb{R}\colon r\mapsto r$ and $g\colon\mathbb{R}\to [-1,1]\colon r\mapsto \frac{2}{\pi}\arctan(r)$. If I refrain from defining any relations, functions, or distinguished elements in the universes of $A$ and $B$, then it is vacuously true that $f$ and $g$ are homomorphisms. Also, $[-1,1]$ and $\mathbb{R}$ would not be isomorphic since $[-1,1]$ has a maximum and minimum element. (Or would this require me to define $\lt$ on $[-1,1]$?).
Are there some other structures, even contrived ones, where such embeddings $f$ and $g$ exist, but $A$ and $B$ are still not isomorphic?