Function symbols in many sorted logic

Question

Consider the following definition

We fix an enumerable set Fun of function symbols. Each function symbol has associated to it an arity of the form $σ1 \times \cdots \times σn \rightarrow σ$, where $n \ge 1$ and $σ1,\cdots , σn, σ$ are sorts. We denote with $Fun_{\{σ1 \times \cdots \times σn \rightarrow σ\}}$ the set of function symbols of arity $σ1 \times \cdots \times σn \rightarrow σ$. We assume that $Fun_{\{σ1 \times \cdots \times σn \rightarrow σ\}}$ is enumerable, for all sorts $σ1, . . . , σn, σ$.

Arity in general refers to the number of arguments a function takes. Is the arity in the definition refers same?

Let $f$ is a function symbol, then is my interpretation true that the domain of $f$ is $σ1 \times \cdots \times σn$ and the range of $f$ is $σ$? If not then what are $σi'$s?

Answer 1

In this case arity is not "only" the number of argument places.

The funcion $f$ has $n$ argument places but each argument place $i$ must be "filled" with a term of sort $\sigma_i$.

Consider e.g a binary function $f$ whose arguments are the first one a natural number and the second one a real and whose value is a complex :

$f : \mathbb N \times \mathbb R \to \mathbb C$.