model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...

, a first-order theory is called model complete if every embedding of its models is an

elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one oft ...

. Equivalently, every first-order formula is equivalent to a universal formula. This notion was introduced by

Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...

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. Not every theory is model-companionable, e.g. theory of groups. However if ''T'' is an

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categorical theory In mathematical logic, a theory is categorical if it has exactly one model ( up to isomorphism). Such a theory can be viewed as ''defining'' its model, uniquely characterizing the model's structure. In first-order logic, only theories with a f ...

, 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 A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age o ...

of ''M'' is

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

. 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 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. Thi ...

. If ''T'' also has universal axiomatization, both of the above are also equivalent to: * ''T''* has

elimination of quantifiers Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that ..." can be viewed as a question "When is there an x such ...

Examples

*Any theory with

is model complete. *The theory of

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...

s is the model completion of the theory of fields. It is model complete but not complete. *The model completion of the theory of

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

s is the theory of equivalence relations with infinitely many equivalence classes, each containing an infinite number of elements. * The theory of

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. Def ...

s, in the language of

ordered ring In abstract algebra, an ordered ring is a (usually 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'' th ...

s, is a model completion of the theory of

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. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...

s (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 field In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field. Alternative definitions The definition given above ...

s, but is not a model completion.

Non-examples

*The theory of dense linear orders with a first and last element is complete but not model complete. *The theory of

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

s (in a language with symbols for the identity, product, and inverses) has the amalgamation property but does not have a model companion.

Sufficient condition for completeness of model-complete theories

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.

Notes

References

* * * * * * {{Authority control Mathematical logic Model theory