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 branch of

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

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

theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...

, 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. Thi ...

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 First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...

and

T

be a theory over

\mathcal L.

For a

model A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...

\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 positive diagram of

\mathfrak A

, sometimes denoted

D^+(\mathfrak A)

, is the set of all those atomic sentences which hold in

\mathfrak A

while the negative diagram, denoted

D^-(\mathfrak A),

thereof is the set of all those atomic sentences which do not hold in

\mathfrak A

. The diagram

D(\mathfrak A)

\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.

Symbolically,

D(\mathfrak A) = D^+(\mathfrak A) \cup \neg D^-(\mathfrak A)

References

Mathematical logic Model theory {{mathlogic-stub

Definition

See also

References