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Mapping of mathematical formulas to a particular meaning, in universal algebra and in model theory

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.

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. 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 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 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. 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.)

Let be again the standard signature for fields. When regarded as structures in the natural way, the rational numbers form a substructure of the real numbers, and the real numbers form a substructure of the complex numbers. The rational numbers are the smallest substructure of the real (or complex) numbers that also satisfies the field axioms. The set of integers gives an even smaller substructure of the real numbers which is not a field. Indeed, the integers are the substructure of the real numbers generated by the empty set, using this signature. The notion in abstract algebra that corresponds to a substructure of a field, in this signature, is that of a subring, rather than that of a subfield. The most obvious way to define a graph is a structure with a signature consisting of a single binary relation symbol The vertices of the graph form the domain of the structure, and for two vertices and means that and are connected by an edge. In this encoding, the notion of induced substructure is more restrictive than the notion of subgraph. For example, let be a graph consisting of two vertices connected by an edge, and let be the graph consisting of the same vertices but no edges. is a subgraph of but not an induced substructure. The notion in graph theory that corresponds to induced substructures is that of induced subgraphs.

As seen above, in the standard encoding of graphs as structures the induced substructures are precisely the induced subgraphs. However, a homomorphism between graphs is the same thing as a homomorphism between the two structures coding the graph. In the example of the previous section, even though the subgraph H of G is not induced, the identity map id: H → G is a homomorphism. This map is in fact a monomorphism in the category σ Hom, and therefore H is a subobject of G which is not an induced substructure.

Structures are sometimes referred to as "first order structures". This is misleading, as nothing in their definition ties them to any specific logic, and in fact they are suitable as semantic objects both for very restricted fragments of first order logic such as that used in universal algebra, and for second order logic. In connection with first order logic and model theory, structures are often called models, even when the question "models of what?" has no obvious answer.

Each first order structure has a satisfaction relation defined for all formulas in the language consisting of the language of together with a constant symbol for each element of which is interpreted as that element. This relation is defined inductively using Tarski's T schema. A structure is said to be a model of a theory if the language of is the same as the language of and every sentence in is satisfied by 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.

A relation is said to be definable with parameters (or definable) if there is a formula with parameters from such that is definable using Every element of a structure is definable using the element itself as a parameter. Some authors use definable to mean definable without parameters, while other authors mean definable with parameters. Broadly speaking, the convention that definable means definable without parameters is more common amongst set theorists, while the opposite convention is more common amongst model theorists.

Structures as defined above are sometimes called one sorted structures to distinguish them from the more general many sorted structures. A many sorted structure can have an arbitrary number of domains. The sorts are part of the signature, and they play the role of names for the different domains. Many sorted signatures also prescribe on which sorts the functions and relations of a many sorted structure are defined. Therefore, the arities of function symbols or relation symbols must be more complicated objects such as tuples of sorts rather than natural numbers. Vector spaces, for example, can be regarded as two sorted structures in the following way. The two sorted signature of vector spaces consists of two sorts V (for vectors) and S (for scalars) and the following function symbols:

+S and ×S of arity (S, S; S). −S of arity (S; S). 0S and 1S of arity (S). +V of arity (V, V; V). −V of arity (V; V). 0V of arity (V). × of arity (S, V; V). If V is a vector space over a field F, the corresponding two sorted structure consists of the vector domain , the scalar domain , and the obvious functions, such as the vector zero , the scalar zero , or scalar multiplication
Many sorted structures are often used as a convenient tool even when they could be avoided with a little effort. But they are rarely defined in a rigorous way, because it is straightforward and tedious (hence unrewarding) to carry out the generalization explicitly. In most mathematical endeavours, not much attention is paid to the sorts. A many sorted logic however naturally leads to a type theory. As Bart Jacobs puts it: "A logic is always a logic over a type theory." This emphasis in turn leads to categorical logic because a logic over a type theory categorically corresponds to one ("total") category, capturing the logic, being fibred over another ("base") category, capturing the type theory.

Both universal algebra and model theory study classes of (structures or) algebras that are defined by a signature and a set of axioms. In the case of model theory these axioms have the form of first order sentences. The formalism of universal algebra is much more restrictive; essentially it only allows first order sentences that have the form of universally quantified equations between terms, e. g.  x y (x + y = y + x). One consequence is that the choice of a signature is more significant in universal algebra than it is in 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. Universal algebra solves this problem by adding a unary function symbol −1. In the case of fields this strategy works only for addition. For multiplication it fails because 0 does not have a multiplicative inverse. An ad hoc attempt to deal with this would be to define 0−1 = 0. (This attempt fails, essentially because with this definition 0 × 0−1 = 1 is not true.) Therefore, one is naturally led to allow partial functions, i. e., functions that are defined only on a subset of their domain. However, there are several obvious ways to generalize notions such as substructure, homomorphism and identity.

In type theory, there are many sorts of variables, each of which has a type. Types are inductively defined; given two types δ and σ there is also a type σ → δ that represents functions from objects of type σ to objects of type δ. A structure for a typed language (in the ordinary first order semantics) must include a separate set of objects of each type, and for a function type the structure must have complete information about the function represented by each object of that type.

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.