Characterizing Completely Multiplicative Polynomial-Arithmetic Functions by Generalized Möbius Functions

Let $Ω$ denote the set of monic polynomials over a finite field and let $A(Ω)$ be the ring of arithmetic functions $f : Ω → C.$ We construct a generalized Möbius functions in $A(Ω)$ and use it to characterize completely multiplicative functions in $A(Ω).$