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Given a first-order theory $T$ its model companion $T^*$ can be considered. Under relatively weak assumptions on $T$ we have that $T^*$ is the theory of the class of the existentially closed models $E(T)$ of $T$. Thus determining the model companion of a theory $T$ of amounts to axiomatizing $E(T)$. Looking at $E(T)$ for the theory of groups and the theory of commutative rings yields that a theory need not have a model companion. When a theory has a model companion the question whether it is finitely axiomatizable is of interest.

I would like to know whether determining $T^*$ and its properties can shed light on $T$ besides being interesting as such. As mentioned above determining $T^*$ amounts to determining $E(T)$, so why not doing so without considering the notion of model companion.

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  • $\begingroup$ Concerning the last paragraph: that's perfectly possible. People have studied existentially closed models even for theories without modal companions (e.g., arithmetic). From a different direction, every theory $T$ has a Kaiser hull $T^{KH}$, which is the largest $\forall\exists$-axiomatized theory with the same universal fragment as $T$. The Kaiser hull is the model companion whenever the latter exists. $\endgroup$ Mar 11, 2015 at 11:41

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From the definition of modelcompanions it follows quickly that (if T is complete) $T^*$ and $T$ will both have the same finite substructures of their models. Because of this reason there have been numerous studies of model companions of classes of structures (their existence and their properties). Since all homogeneous structures are genereated, by the classic construction of Fraïssé, from their set of finite substructures, these model companions are extra nice, though getting information in the other direction (which you ask about) is harder. There is an old result by Saracino that if $T$ is $\omega$-categorical then $T^*$ is unique and also $\omega$-categorical. Thus if you start with a non $\omega$-categorical $T^*$, we conclude that $T$ also is not $\omega$-categorical.

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