TITLE: Elementary embedding and elementary equivalence
QUESTION [5 upvotes]: I guess it is not difficult, but I spent an hour thinking about this without success.
Elementary embedding implies elementary equivalence, but elementary equivalence between structures does not implies, necessarily, the existence of an elementary embedding between these structures. 
I`m looking for an example of two structures that are elementary equivalent but there is no elementary embedding between them.
any ideas?
Thanks!

REPLY [6 votes]: Consider the theory of equivalence relations with exactly two infinite equivalence classes, which is a complete theory, so every pair of models is elementarily equivalent. Let $M$ be a model with classes of size $\aleph_0$ and $\aleph_2$ and let $N$ be a model with classes of size $\aleph_1$ and $\aleph_1$. Then $M$ cannot embed in $N$ because a class of size $\aleph_2$ cannot embed in a class of size $\aleph_1$, and $N$ cannot embed in $M$ for a similar reason. So there is no embedding at all, much less an elementary embedding.