In the mathematical discipline of model theory, a structure $\backslash mathfrak\{A\}$ for a language $\backslash mathcal\{L\}$ (referred to as an '$\backslash mathcal\{L\}$-structure', and commonly written as a Gothic capital) is an ordered pair whose first member is the domain of discourse or universe set $\backslash mathit\{A\}$ (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 $\backslash mathcal\{I\}$, i.e. a partial function of $\backslash mathcal\{L\}$ which is defined on precisely the non-logical symbols of $\backslash mathcal\{L\}$ so that the constant symbols of $\backslash mathcal\{L\}$, if any, are taken to elements of $\backslash mathit\{A\}$, the function symbols of $\backslash mathcal\{L\}$, if any, are taken to functions on $\backslash mathit\{A\}$, and the relation symbols of $\backslash mathcal\{L\}$, if any, are taken to relations on $\backslash mathit\{A\}$.

Usage note

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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.

Model theory