On a class of \(\psi\)-products preserving multiplicativity. II (Q1345192)

Let \(T\) be a non-empty subset of \(\mathbb{N}\times \mathbb{N}\) and \(\psi: T\to \mathbb{N}\) a mapping satisfying conditions concerning finiteness of solutions, association and symmetry. The \(\psi\)-product of arithmetic functions \(f\), \(g\) is defined by \[ (f\psi g) (n)= \sum_{\psi (x, y)=n} f(x) g(y). \] \(\psi\) is called multiplicativity preserving if the \(\psi\)- product of multiplicative functions is again multiplicative. The authors widen the class of \(\psi\)-functions investigated in their earlier paper [Indian J. Pure Appl. Math. 22, 819-832 (1991; Zbl 0751.11006)]\ by characterizing the multiplicativity preserving functions \(\psi\) which are onto. Further they show that the arithmetic functions \(f\) which are invertible with respect to \(\psi\) form a group if \(\psi(x, y)\geq \max\{ x, y\}\) for all \((x, y)\in T\).