Model complete theory

In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula. This notion was introduced by Abraham Robinson.

Model companion and model completion
A companion of a theory T is a  theory T* such that every model of T can be embedded in a model of  T* and vice versa.

A model companion of a theory T is a companion of T that is model complete. Robinson proved that a theory has at most one model companion. Not every theory is model-companionable, e.g. theory of groups. However if T is an $$\aleph_0$$-categorical theory, then it always has a model companion.

A model completion for a theory T is a model companion T* such that for any model M of T, the theory of T* together with the diagram of M is complete. Roughly speaking, this means every model of T is embeddable in a model of T* in a unique way.

If T* is a model companion of T  then the following conditions are equivalent:
 * T* is a model completion of T
 * T has the amalgamation property.

If T also has universal axiomatization, both of the above are also equivalent to:
 * T* has elimination of quantifiers

Examples

 * Any theory with elimination of quantifiers is model complete.
 * The theory of algebraically closed fields is the model completion of the theory of fields. It is model complete but not complete.
 * The model completion of the theory of equivalence relations is the theory of equivalence relations with infinitely many equivalence classes, each containing an infinite number of elements.
 * The theory of real closed fields, in the language of ordered rings, is a model completion of the theory of ordered fields (or even ordered domains).
 * The theory of real closed fields, in the language of rings, is the model companion for the theory of formally real fields, but is not a model completion.

Non-examples

 * The theory of dense linear orders with a first and last element is complete but not model complete.
 * The theory of groups (in a language with symbols for the identity, product, and inverses) has the amalgamation property but does not have a model companion.

Sufficient condition for completeness of model-complete theories
If T is a model complete theory and there is a model of T that embeds into any model of T, then T is complete.