Model complete theory

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 -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

• 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.

• 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.

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.