Structure/Model for first order language

Question

For simplicity lets say $L$ is a first order language with a single function symbol $f$ and no other nonlogical symbols

What would a structure/model, $M$ say, for $L$ look like? I know $M$ has some non-empty domain but Im confused about whether $M$ would contain both the symbol $f$ and a new symbol $f^m$ for the interpretation of f in M or would it just contain the interpretation symbol. Which of these is in the domain?

Thanks

Answer 1

A structure in this language would consist of

• a domain $D$, and
• a function $D\to D$ that is the interpretation of $f$.

If you want to be completely formal of it, the structure would probably have the domain, then a map that maps the symbol $f$ to the function that interprets it, and then the empty map (for mapping the non-existent constant symbols of the language to their interpretations). But these nitty-gritty details depend a lot on how exactly your textbook/lecturer has chosen to set up things, and they are not really important.