I never quite got the difference between an "interpretation" and a "structure" in logic. To me, they always sound like the same thing: a function from elements of a language to a particular universe.
The "Structure (mathematical logic)" page on Wikipedia says:
Formally, a structure can be defined as a triple $\mathcal A=(A, \sigma, I)$ consisting of a domain $A$, a signature $\sigma$, and an interpretation function $I$ that indicates how the signature is to be interpreted on the domain.
From that, an interpretation is merely a component of a structure. However, I would say the function $I$ necessarily contains information about its domain (as in "function domain", a different sense of "domain" as used in the quote above) and its codomain, which must be $\sigma$ and $A$, respectively, so there is always a one-to-one correspondence between structures and interpretations.
Can anyone clarify what I am missing here? Thanks.