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I should state first that this reply has only to do with the above mentioned ideas in the category of models of a first order theory.

John Goodrick's work is referenced in Joel's post above, and I have heard John Goodrick speak about this at least once. Specifically, John mentioned the following (and a lot more that I didn't write down):

Fix some countable, complete first-order theory, $T$. Suppose $T$ has the following property: Whenever we are given two models $\mathcal M_1$ and $\mathcal M_2$ of $T$ which have elementary embeddings into each other, then $\mathcal M_1 \cong \mathcal M_2.$

Then $T$ is superstable and nonmultidimensional (and I know if John replies to this, he can mention many other things, but I don't remember now). In the case that $T$ is actually $\omega -$stable, nonmultidimensional implies the bi-embedding property stated in the above paragraph.

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John Goodrick's work is referenced in Joel's post above, and I have heard John Goodrick speak about this at least once. Specifically, John mentioned the following (and a lot more that I didn't write down):

Fix some countable, complete first-order theory, $T$. Suppose $T$ has the following property: Whenever we are given two models $\mathcal M_1$ and $\mathcal M_2$ of $T$ which have elementary embeddings into each other, then $\mathcal M_1 \cong \mathcal M_2.$

Then $T$ is superstable and nonmultidimensional (and I know if John replies to this, he can mention many other things, but I don't remember now). In the case that $T$ is actually $\omega -$stable, nonmultidimensional implies the bi-embedding property stated in the above paragraph.
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