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| Contents |
|---|
| Definition |
| Domain |
| Induced substructures and closed subsets |
| Examples |
| Homomorphisms |
| Embeddings |
| Definable relations |
| Structures that are proper classes |
| See also |
| Notes |
| External links |
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.
Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures of first-order theories with no relation symbols. Model theory has a different scope that encompasses more arbitrary first-order theories, including foundational structures such as models of set theory.
From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic, cf. also Tarski's theory of truth or Tarskian semantics.
For a given theory in model theory, a structure is called a model if it satisfies all the sentences of that theory. Logicians sometimes refer to structures as "interpretations", whereas the term "interpretation" generally has a different (although related) meaning in model theory; see interpretation (model theory).
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In the context of mathematical logic, the term "model" was first used in 1940 by the philosopher Willard Van Orman Quine, in a reference to mathematician Richard Dedekind (1831–1916), a pioneer in the development of set theory. The term "theory of models" was coined by Alfred Tarski, a member of the Lwów–Warsaw school, in 1954.
Since the 19th century, one main method for proving the consistency of a set of axioms has been to provide a model for it.
Formally, a structure can be defined as a triple σ, and an interpretation function σ-structure.
The natural number s because it is the arity of the interpretation
The interpretation function f of arity f𝒜=I(f) on the domain. Each relation symbol n-ary relation c is called a constant symbol, because its interpretation I(s). For example, if f:𝒜2→𝒜 rather than + and +) and the two constant symbols + and ℚ, the real numbers σ-structures in an obvious way: Sf={+,×,−,0,1} and I𝒬(+):ℚ×ℚ→ℚ is addition of rational numbers,
𝒜 and 𝒜 is contained in the domain of |𝒜|.
The usual notation for this relation is 𝒜 is called closed if it is closed under the functions of n-ary function symbol b1,b2,…,bn∈B, the result of applying b1b2…bn is again an element of B⊆|𝒜| there is a smallest closed subset of B, or the hull of ⟨B⟩𝒜. The operator 𝒜=(A,σ,I) and 𝒜, where 𝒜. Conversely, the domain of an induced substructure is a closed subset.
The closed subsets (or induced substructures) of a structure form a lattice. The meet of two subsets is their intersection. The join of two subsets is the closed subset generated by their union. Universal algebra studies the lattice of substructures of a structure in detail.
Let σ consisting of a single binary relation symbol b, b are connected by an edge. In this encoding, the notion of induced substructure is more restrictive than the notion of subgraph. For example, let H is a subgraph of
Given two structures 𝒜 to a1,a2,…,an∈|𝒜|, the following equation holds:
where R in the structure 𝒜 to |𝒜|, ℬ.
For every signature σ there is a concrete category σ-Hom which has σ-structures as objects and σ-homomorphisms as morphisms.
A homomorphism (h(a1),h(a2),…,h(an))∈Rℬ, then there are h(a1′)=h(a1),h(a2′)=h(a2),…,h(an′)=h(an).
The strong homomorphisms give rise to a subcategory of the category σ-Hom that was defined above.
A (σ-)homomorphism (a1,a2,…,an)∈R𝒜⟺(h(a1),h(a2),…,h(an))∈Rℬ
(where R in the structure 𝒜 and ℳ⊨ϕ defined for all formulas M, which is interpreted as that element. This relation is defined inductively using Tarski's T-schema.
A structure ℳ is the same as the language of ℳ. Thus, for example, a "ring" is a structure for the language of rings that satisfies each of the ring axioms, and a model of ZFC set theory is a structure in the language of set theory that satisfies each of the ZFC axioms.
An M of the structure ∅︀ cf. below) if there is a formula R is definable if and only if there is a formula m of φ(x) such that |ℳ|-definable) if there is a formula ℳ such that while other authors mean definable with parameters.R on the universe φ(x1,…,xn) such that R must be over the signature of R itself, since φ in the extended language containing the language of R is the only relation on R is said to be implicitly definable over 𝒱 consists of the vector domain 0V𝒱=0∈|𝒱|V, the scalar zero ∀ x
There is more than one possible semantics for higher-order logic, as discussed in the article on second-order logic. When using full higher-order semantics, a structure need only have a universe for objects of type 0, and the T-schema is extended so that a quantifier over a higher-order type is satisfied by the model if and only if it is disquotationally true. When using first-order semantics, an additional sort is added for each higher-order type, as in the case of a many sorted first order language.
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. These structures are sometimes called class models to distinguish them from the "set models" discussed above. When the domain is a proper class, each function and relation symbol may also be represented by a proper class.
In Bertrand Russell's Principia Mathematica, structures were also allowed to have a proper class as their domain.
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