In the mathematical discipline of model theory, a structure for a language (referred to as an '-structure', and commonly written as a Gothic capital) is an ordered pair whose first member is the domain of discourse or universe set A (taken to be a set with possibly relations and functions defined on it, and commonly written as the Roman capital corresponding to the name of the structure) and whose second member is an interpretation , i.e. a partial function of which is defined on precisely the non-logical symbols of so that the constant symbols of , if any, are taken to elements of A, the function symbols of , if any, are taken to functions on A, and the relation symbols of , if any, are taken to relations on A. In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ... The domain of discourse, sometimes called the universe of discourse, is an analytic tool used in deductive logic, especially predicate logic. ... In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ... Look up Relation in Wiktionary, the free dictionary In mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation (of set theory and logic) and relational algebra. ... Look up Function in Wiktionary, the free dictionary The word function may mean: In common parlance, a role of a component in an assembly, or of an element in a systemic aggregate (such as a person within a group). ... Interpretation, or interpreting, is an activity that consists of establishing, either simultaneously or consecutively, oral or gestural communications between two or more speakers who are not speaking (or signing) the same language. ...

Usage note

The term model, as used in model theory, is essentially synonymous with "structure", but tends to be used in different contexts. Typically, the term "model" is used when one has a specific theory in mind, and is considering only models of that theory—that is, structures that satisfy every sentence in the theory. "Structure", on the other hand, tends to be used when less of the behavior of the structure is known or specified. In mathematical logic, a theory is usually defined as a set of first-order sentences (closed first-order formulas). ...