Model complete theory

Remarks

Equivalently, every embedding between models of $T$ is elementary. By Robinson's test, $T$ is model complete if and only if every $L$-formula is equivalent modulo $T$ to an existential formula $\exists \bar{y} \, \chi(\bar{x}, \bar{y})$ with $\chi$ quantifier-free. A theory with quantifier elimination is model complete.

Example: the theory of dense linear orders without endpoints is model complete, since it has quantifier elimination. Let $M \subseteq N$ be two such orders and let $\bar{a}$ be a tuple of $M$. A formula $\varphi(\bar{a})$ is equivalent to a quantifier-free formula. That formula only compares elements of $\bar{a}$, so it has the same truth value in $M$ and in $N$.

Non-example: the theory of dense linear orders with a least and a greatest element is not model complete. The orders $[0, 1] \cap \Q$ and $[0, 2] \cap \Q$ are models, and the first is a substructure of the second. The element $1$ satisfies $\forall y \, (y \le x)$ in $[0, 1] \cap \Q$ but not in $[0, 2] \cap \Q$.

The theory of real closed fields in the signature $\set{+, -, \cdot, 0, 1}$ is model complete but does not have quantifier elimination.