Structure (mathematical logic)

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

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.

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