I am currently reading Logic of Mathematics by Sofia Adamowicz and Paweł Zbierski, and the following definition is given
$$\textrm{arg}(r)=n \textrm{ if and only if } r\subset A^n$$ however, it seems to follow then that the argument function is not well-defined for the empty set.
Likewise
$$\textrm{arg}(f)=n \textrm{ if and only if dom}(f)= A^n$$ and this also seems not well-defined for the empty function. Is there a canonical way to handle this. I ask because it affects the answer I give to one of the problems in the textbook, and I'd prefer to have the `right' one.