This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (April 2010) (Learn how and when to remove this message)

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.^[1] 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 the defining axioms of that theory, although it is sometimes disambiguated as a semantic model when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as "interpretations",^[2] whereas the term "interpretation" generally has a different (although related) meaning in model theory, see interpretation (model theory).

In database theory, structures with no functions are studied as models for relational databases, in the form of relational models.

History

This section needs expansion with: explicit mention of the term "structure". You can help by adding to it. (November 2023)

In the context of mathematical logic, the term "model" was first applied 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.^[3][4] Since the 19th century, one main method for proving the consistency of a set of axioms has been to provide a model for it.

Definition

Formally, a structure can be defined as a triple ${\displaystyle {\mathcal {A}}=(A,\sigma ,I)}$ consisting of a domain ${\displaystyle A,}$ a signature ${\displaystyle \sigma ,}$ and an interpretation function ${\displaystyle I}$ that indicates how the signature is to be interpreted on the domain. To indicate that a structure has a particular signature ${\displaystyle \sigma }$ one can refer to it as a ${\displaystyle \sigma }$-structure.

Domain

The domain of a structure is an arbitrary set; it is also called the underlying set of the structure, its carrier (especially in universal algebra), its universe (especially in model theory, cf. universe), or its domain of discourse. In classical first-order logic, the definition of a structure prohibits the empty domain.^{[citation needed][5]}

Sometimes the notation ${\displaystyle \operatorname {dom} ({\mathcal {A}})}$ or ${\displaystyle |{\mathcal {A}}|}$ is used for the domain of ${\displaystyle {\mathcal {A}},}$ but often no notational distinction is made between a structure and its domain (that is, the same symbol ${\displaystyle {\mathcal {A}}}$ refers both to the structure and its domain.)^[6]

Signature

The signature ${\displaystyle \sigma =(S,\operatorname {ar} )}$ of a structure consists of:

• a set ${\displaystyle S}$ of function symbols and relation symbols, along with
• a function ${\displaystyle \operatorname {ar} :\ S\to \mathbb {N} _{0}}$ that ascribes to each symbol ${\displaystyle s}$ a natural number ${\displaystyle n=\operatorname {ar} (s).}$

The natural number ${\displaystyle n=\operatorname {ar} (s)}$ of a symbol ${\displaystyle s}$ is called the arity of ${\displaystyle s}$ because it is the arity of the interpretation^{[clarification needed]} of ${\displaystyle s.}$

Since the signatures that arise in algebra often contain only function symbols, a signature with no relation symbols is called an algebraic signature. A structure with such a signature is also called an algebra; this should not be confused with the notion of an algebra over a field.

Interpretation function

The interpretation function ${\displaystyle I}$ of ${\displaystyle {\mathcal {A}}}$ assigns functions and relations to the symbols of the signature. To each function symbol ${\displaystyle f}$ of arity ${\displaystyle n}$ is assigned an ${\displaystyle n}$-ary function ${\displaystyle f^{\mathcal {A}}=I(f)}$ on the domain. Each relation symbol ${\displaystyle R}$ of arity ${\displaystyle n}$ is assigned an ${\displaystyle n}$-ary relation ${\displaystyle R^{\mathcal {A}}=I(R)\subseteq A^{\operatorname {ar(R)} }}$ on the domain. A nullary (${\displaystyle =\,0}$-ary) function symbol ${\displaystyle c}$ is called a constant symbol, because its interpretation ${\displaystyle I(c)}$ can be identified with a constant element of the domain.

When a structure (and hence an interpretation function) is given by context, no notational distinction is made between a symbol ${\displaystyle s}$ and its interpretation ${\displaystyle I(s).}$ For example, if ${\displaystyle f}$ is a binary function symbol of ${\displaystyle {\mathcal {A}},}$ one simply writes ${\displaystyle f:{\mathcal {A}}^{2}\to {\mathcal {A}}}$ rather than $$Zdroj:https://en.wikipedia.org?pojem=Interpretation_function
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.