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Finitely generated substructures and/or local finiteness seem to play a great role in the study of model completions. For instance, I am told the following:

A universal theory $T$ has a model completion $T^*$ iff every e.c. model of $T$ satisfies $T^*$ and for every $M\models T^*$, $M_0 \subseteq M$ finitely generated, and $M_0 \subseteq M_1 \models T$, there is an embedding from $M_1$ to an elementary extension of $M$ that preserves $M_0$ pointwise.

While it is easy to prove this, I wonder if there is anything about finitely generated substructures and model completions. Is there a good source from which I can learn similar facts? Obvious sources like the relevant chapter in the Handbook of Mathematical Logic, Chang & Keisler, the larger Hodges, or Marker was not helpful.

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  • $\begingroup$ "e.c." is ? equationnally compact ? then is $T$ only in a functional language ?(that's what the "universal algebra" tag suggests but...) $\endgroup$ – Maxime Ramzi Dec 4 '17 at 22:00
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    $\begingroup$ @Max existentially closed $\endgroup$ – Alex Kruckman Dec 4 '17 at 23:37
  • $\begingroup$ @AlexKruckman : Thanks, I was completely mistaken ! $\endgroup$ – Maxime Ramzi Dec 5 '17 at 7:11
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The theorem you cite is important because it tells us that in order to prove whether or not a theory T’ is a model completion, it is sufficient to check a certain condition on the finitely generated substructures of models of T’. This can be a useful reduction. However, in general I do not think finitely generated substructures of models of a model companion hold any special sort of structure that model theorists have taken the time to study in the abstract. Typically the interest in “finitely generated substructures” in statements like these is just the fact that we can work with finitely many parameters at a time.

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