A Dirichlet analogue of the free monogenic inverse semigroup via Möbius inversion. (Q640282)

A Möbius category \(C\) is a small category in which morphisms have only finitely many non-trivial factorizations and any complex-valued function defined on \(\text{Mor\,}C\) has a convolution inverse if and only if for each identity morphism \(\xi\) one has \(f(\xi)\neq 0\). The Möbius function of \(C\) is the convolution inverse of the constant function \(f(\xi)=1\). The authors consider the Möbius category of integer affine maps \(ax+b\) with integral \(a>0,b\), determine its Möbius function, construct its Leech inverse monoid \(I_D\) and obtain the Möbius inversion formula for \(I_D\).