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==