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, 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.
- 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.
Read more about this topic: Model Complete Theory
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