Structure (mathematical logic)

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 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", 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. Formally, a structure can be defined as a triple consisting of a domain a signature and an interpretation function that indicates how the signature is to be interpreted on the domain. To indicate that a structure has a particular signature one can refer to it as a -structure. The domain of a structure is an arbitrary set; it is also called the of the structure, its (especially in universal algebra), its (especially in model theory, cf. universe), or its . In classical first-order logic, the definition of a structure prohibits the empty domain. Sometimes the notation or is used for the domain of but often no notational distinction is made between a structure and its domain (that is, the same symbol refers both to the structure and its domain.