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For the sake of having a reference to link to later, I’ve written diagram of a first-order structure. This is just a construction where you take a theory $T$ and expand it to a new theory $T'$ by naming one of its models $M$ with constant symbols for each element of $M$ while additionally stipulating those constant symbols have to behave like they came from $M$.
If those additional stipulations were only quantifier-free, the models of $T'$ are those models of $T$ containing $M$ as an induced substructure.
If those additional stipulations were all the first-order sentences satisfied by the elements of $M$, then the models of $T'$ are those models of $T$ containing $M$ as an elementary substructure.
I added a remark (currently Remark 4.1) to diagram of a first-order structure that passing from $T$ to $T_{\mathsf{Diag(M)}}$ (resp. $\mathsf{EDiag}$) is functorial on interpretations the way that one would expect.
Jesse,
with all the nice material that you are adding on model theory. Might you have the energy to expand the (presently puny) table of contents at
?
(This entry is the one that is included as the “floating table of contents” in the top right of all the model theory entries.)
As an example for what I have in mind, compare to
Thanks for the reminder! It’s now on my to-do list.
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