Structure (mathematical logic)
Interpretation function

The interpretation function I {\displaystyle I} of A {\displaystyle {\mathcal {A}}} assigns functions and relations to the symbols of the signature. To each function symbol f {\displaystyle f} of arity n {\displaystyle n} is assigned an n {\displaystyle n} -ary function f A = I (f) {\displaystyle f^{\mathcal {A}}=I(f)} on the domain. Each relation symbol R {\displaystyle R} of arity n {\displaystyle n} is assigned an n {\displaystyle n} -ary relation R A = I (R) ⊆ A a r (R) {\displaystyle R^{\mathcal {A}}=I(R)\subseteq A^{\operatorname {ar(R)} }} on the domain. A nullary (= 0 {\displaystyle =\,0} -ary) function symbol c {\displaystyle c} is called a constant symbol, because its interpretation I (c) {\displaystyle I(c)} can be identified with a constant element of the domain.

When a structure (and hence an interpretation function) is given by context, no notational distinction is made between a symbol s {\displaystyle s} and its interpretation I (s). {\displaystyle I(s).} For example, if f {\displaystyle f} is a binary function symbol of A, {\displaystyle {\mathcal {A}},} one simply writes f: A 2 → A {\displaystyle f:{\mathcal {A}}^{2}\to {\mathcal {A}}} rather than f A: | A | 2 → | A |. {\displaystyle f^{\mathcal {A}}:|{\mathcal {A}}|^{2}\to |{\mathcal {A}}|.}
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