Model complete theory

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In model theory, a theory is called model complete if every embedding of models is an elementary equivalence. This notion was introduced by Abraham Robinson.

[edit] 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.

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:

[edit] Examples

  • The theory of dense linear orders with a first and last element is complete but not model complete.
  • The theory of dense linear orders with two constant symbols is model complete but not complete.
  • The theory of algebraically closed fields is the model completion of the theory of fields. It is model complete but not complete.
  • The theory of real closed fields is the model companion for the theory of formally real fields, but is not a model completion.
  • Any theory with elimination of quantifiers is model complete.
  • The model completion of the theory of equivalence relations is the theory of equivalence relations with infinitely many equivalence classes.
  • 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.

[edit] References