In model theory

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

, a first-order

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

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

Abraham Robinson

Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....

.

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:

• T* is a model completion of T
• T* has elimination of quantifiers
• T has the amalgamation property
  Amalgamation property
  In the mathematical field of model theory, the amalgamation property is a property of collections of structures that guarantees, under certain conditions, that two structures in the collection can be regarded as substructures of a larger one....
  
  .

Examples

• The theory of dense linear orders with a first and last element is complete
  Complete theory
  In mathematical logic, a theory is complete if it is a maximal consistent set of sentences, i.e., if it is consistent, and none of its proper extensions is consistent...
  
   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 field
  Algebraically closed field
  In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
  
  s is the model completion of the theory of fields. It is model complete but not complete.
• The theory of real closed field
  Real closed field
  In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.-Definitions:...
  
  s, in the language of ordered ring
  Ordered ring
  In abstract algebra, an ordered ring is a commutative ring R with a total order ≤ such that for all a, b, and c in R:* if a ≤ b then a + c ≤ b + c.* if 0 ≤ a and 0 ≤ b then 0 ≤ ab....
  
  s, is a model completion of the theory of ordered field
  Ordered field
  In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
  
  s (or even ordered domains). The theory of real closed fields, in the language of rings
  Ring (mathematics)
  In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
  
  , is the model companion for the theory of formally real field
  Formally real field
  In mathematics, in particular in field theory and real algebra, a formally real field is a field that admits an ordering which makes it an ordered field.-Alternative Definitions:...
  
  s, but is not a model completion.
• Any theory with elimination of quantifiers is model complete.
• The model completion of the theory of equivalence relation
  Equivalence relation
  In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
  
  s is the theory of equivalence relations with infinitely many equivalence classes.
• The theory of group
  Group (mathematics)
  In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
  
  s (in a language with symbols for the identity, product, and inverses) has the amalgamation property but does not have a model companion.