What do $n$ and $k$ refer to when we have a function symbol $f^n_k$?

Question

What do $n$ and $k$ refer to when we have a function symbol $f^n_k$?

From what I undesrtand, $n$ is the arity of the function, for example we can write the product function $a * b$ as $f^2_1 a b$

I remember my teacher saying something about $k$ being "k many functions". What does this mean?

Answer 1

I am assuming the context is a presentation of first-order predicate calculus. In that case, $k$ is just a number that uniquely identifies $f^n_k$ among the various function symbols of arity $n$ in the theory of interest. So in the theory of a ring, you would have two binary operators (addition and multiplication) and you might choose to use $f^2_1$ for addition and $f^2_2$ for multiplication.

Answer 2

That notation shows up in many contexts with different meaning.

From the little bit of context you display it could indeed be $$ f_k^n : A^n \to B $$ As you can evaluate a function of $n$ arguments in $n$-steps, using Schönfinkel/Frege/Curry you can associate it with $n$ functions of a single argument each.