Structure (mathematical logic)

structuremodelstructuresmodelsrelational structurealgebraInterpretation functionsatisfaction relationsorts structure
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.wikipedia
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Universal algebra

algebraequational theoryequational reasoning
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.
In universal algebra, an algebra (or algebraic structure) is a set A together with a collection of operations on A.

Model theory

modelmodelsmodel-theoretic
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it. Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory.
In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

Algebraic structure

algebraic structuresunderlying setalgebraic system
Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces.
In the context of universal algebra, the set A with this structure is called an algebra, while, in other contexts, it is (somewhat ambiguously) called an algebraic structure, the term algebra being reserved for specific algebraic structures that are vector spaces over a field or modules over a commutative ring.

Relational model

relationalrelational data modelrelationships
In database theory, structures with no functions are studied as models for relational databases, in the form of relational models.
The content of the database at any given time is a finite (logical) model of the database, i.e. a set of relations, one per predicate variable, such that all predicates are satisfied.

Interpretation (logic)

interpretationintended interpretationinterpretations
Logicians sometimes refer to structures as interpretations.
If a given interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentence or theory.

Foundations of mathematics

foundation of mathematicsfoundationsfoundational
Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory.
At that time, the main method for proving the consistency of a set of axioms was to provide a model for it.

Substructure (mathematics)

substructureextensionsubalgebra
In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are the traces of the functions and relations of the bigger structure.

Closure operator

interior operatorclosurekernel operator
The operator is a finitary closure operator on the set of subsets of.
Every subset of an algebra generates a subalgebra: the smallest subalgebra containing the set.

Finite model theory

finite modelsfinite model
Therefore, the [[Complexity of constraint satisfaction#Constraint satisfaction and the homomorphism problem|complexity of CSP]] can be studied using the methods of finite model theory.
FMT is a restriction of MT to interpretations on finite structures, which have a finite universe.

Signature (logic)

signaturesignatureslanguage
Formally, a structure can be defined as a triple consisting of a domain A, a signature σ, and an interpretation function I that indicates how the signature is to be interpreted on the domain.
In a structure, an interpretation ties the function and relation symbols to mathematical objects that justify their names: The interpretation of an n-ary function symbol f in a structure A with domain A is a function f A : A n → A, and the interpretation of an n-ary relation symbol is a relation R A ⊆ A n .

Map (mathematics)

mappingmapmaps
Given two structures \mathcal A and \mathcal B of the same signature σ, a homomorphism from \mathcal A to \mathcal B is a map that preserves the functions and relations.
In formal logic, the term map is sometimes used for a functional predicate, whereas a function is a model of such a predicate in set theory.

Theory (mathematical logic)

theorytheoriesformal theories
A structure \mathcal{M} is said to be a model of a theory T if the language of \mathcal{M} is the same as the language of T and every sentence in T is satisfied by \mathcal{M}.
A satisfiable theory is a theory that has a model.

Elementary class

axiomatizabilityAxiomatizable classmodule (model theory)
For example, the class of groups, in the signature consisting of the binary function symbol × and the constant symbol 1, is an elementary class, but it is not a variety.
In model theory, a branch of mathematical logic, an elementary class (or axiomatizable class) is a class consisting of all structures satisfying a fixed first-order theory.

Graph homomorphism

homomorphismcategory of graphsgraph homomorphisms
However, a homomorphism between graphs is the same thing as a homomorphism between the two structures coding the graph.
Graphs and directed graphs can be viewed as a special case of the far more general notion called relational structures (defined as a set with a tuple of relations on it).

Many-sorted logic

sortsmany-sortedOrder-sorted logic
A many-sorted logic however naturally leads to a type theory.

Graph (discrete mathematics)

graphundirected graphgraphs
The most obvious way to define a graph is a structure with a signature σ consisting of a single binary relation symbol E.
In model theory, a graph is just a structure.

Concrete category

concrete categoriesconcreteconcretizable
For every signature σ there is a concrete category σ-Hom which has σ-structures as objects and σ-homomorphisms as morphisms.
For example, it may be useful to think of the models of a theory with N sorts as forming a concrete category over Set N.

Class (set theory)

classproper classclasses
In the study of set theory and category theory, it is sometimes useful to consider structures in which the domain of discourse is a proper class instead of a set.
Semantically, in a metalanguage, the classes can be described as equivalence classes of logical formulas: If \mathcal A is a structure interpreting ZF, then the object language class builder expression is interpreted in \mathcal A by the collection of all the elements from the domain of \mathcal A on which holds; thus, the class can be described as the set of all predicates equivalent to \phi (including \phi itself).

Set (mathematics)

setsetsmathematical set
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.

Finitary

infinitaryfinitary logicfinitary operations
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.

Finitary relation

relationrelationsTheory of relations
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.

Group (mathematics)

groupgroupsgroup operation
Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces.

Ring (mathematics)

ringringsassociative ring
Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces.

Field (mathematics)

fieldfieldsfield theory
Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces.

Vector space

vectorvector spacesvectors
Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces.