model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...

, a branch of

mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...

, the diagram of a structure is a simple but powerful concept for proving useful properties of a

theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...

, for example 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. This ...

and the

joint embedding property In universal algebra and model theory, a class of structures ''K'' is said to have the joint embedding property if for all structures ''A'' and ''B'' in ''K'', there is a structure ''C'' in ''K'' such that both ''A'' and ''B'' have embeddings into ...

, among others.

Definition

Let

\mathcal L

be a first-order language and

T

be a theory over

\mathcal L.

For a model

\mathfrak A

T

one expands

\mathcal L

to a new language :

\mathcal L_A := \mathcal L\cup \

by adding a new constant symbol

c_a

for each element

a

A,

where

A

is a subset of the domain of

\mathfrak A.

Now one may expand

\mathfrak A

to the model :

\mathfrak A_A := (\mathfrak A,a)_.

The diagram of

\mathfrak A

is the set of all

atomic sentence In logic and analytic philosophy, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences. ...

s and negations of atomic sentences of

\mathcal L_A

that hold in

\mathfrak A_A.

References

{{mathlogic-stub Mathematical logic Model theory

Definition

See also

References