Axiomatize the class of existentially closed structures

Question

Let $\mathcal{T}$ be a theory in a language $\mathcal{L}$. A model $M$ of $\mathcal{T}$ is existentially-closed if for every existential $\mathcal{L}$-formula $\varphi(\bar{x})$, every extension $N \supseteq M$, and tuple $\bar{m}\in M$ we have that $N\models \varphi(\bar{m})$ implies $M\models \varphi(\bar{m})$. I know that if $\mathcal{T}$ is inductive (i.e., the class of its models is closed under unions of embedding chains) every model of $\mathcal{T}$ can be extended to an existentially-closed model.

In the examples I've seen in class there is always a theory $\mathcal{T}^{ec}$ that axiomatizes existentially-closed models of $\mathcal{T}$ (e.g. algebraically closed fields, real closed fields, DLOs without endpoints, etc.)

Q1: Does $\mathcal{T}^{ec}$ always exist? Do we need to assume that $\mathcal{T}$ is inductive?

If the answer is positive then this would imply

P: Any structure elementarily equivalent to an existentially closed model would itself be an existentially closed model.

This leads me to my second question

Q2: If the answer to Q1 is negative, can I have some counterexamples to $P$?

Answer 1

For inductive theories $T$, $T$ has a model companion if and only if the class of existentially closed models of $T$ is elementary (axiomatized by a first-order theory). An inductive theory satisfying these equivalent conditions is called companionable. If $T$ is companionable, then the theory $T^{ec}$ of its existentially closed models is the unique model companion of $T$. See Section 8.3 of Model Theory by Hodges.

Some examples of inductive theories which are not companionable:

• The theory of groups. See Exercise 8.5.2 on p. 398 in Hodges.
• The theory of commutative rings. See Example 1 on p. 393 in Hodges.
• The theory of acyclic graphs (i.e., forests). See here.
• Let $T$ be an inductive $L$-theory, and let $T_\sigma$ be the theory obtained by adding a new unary function symbol $\sigma$ and axioms asserting that $\sigma$ is an $L$-automorphism. If $T_\sigma$ is companionable, we call the model companion "$T$ with a generic automorphism". If $T$ has the strict order property (e.g. if models of $T$ admit a definable linear ordering), then $T_\sigma$ is not companionable. See this paper by Kikyo and Shelah. For theories without the strict order property, the conditions under which $T_\sigma$ is companionable are extremely complicated. See, for example, this paper by Baldwin and Shelah.