Structure (mathematical logic)

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

Thus, $\mathcal{I}$ is a function from non-logical symbols to their extension, respecting signature.

By splitting off the notion of signature, we can build a slightly other (maybe more economical) approach. By this, we can e.g. take a first-order language $\mathcal L$ and a structure $\mathfrak A$ , both of a common signature $Σ$ . This approach may affect the definition of notions like assignment slightly. Because the former unidirectional dependence between first-order language $\mathcal L$ and an $\mathcal L$ -structure $\mathfrak A$ is replaced now by a V-shaped dependence: it may be more economical to build a dependence of both language and structure on a common underlying concept called signature.

[edit] 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.