The Möbius inverse formulas on Abelian semigroups

Let \(\Lambda\) be a commutative ring with identity element. Given a locally finite Abelian semigroup $X$ with identity element ${1\mspace{-4.85mu}{\mathrm I}}$ one may ask if the Möbius–type \(\Lambda\)–valued function exists on $X$. As it is proved in the present paper the existence of such a function often depends on the following property of $\zeta$–function of $X$: this function has not zeros $\chi$ such that the support of the character $\chi$ is a finite subset of $X$. \(\mathbb{Z}\)–valued Möbius function exists if and only if \(x^2=x\) implies \(x={1\mspace{-4.85mu}{\mathrm I}}\). Bibl. 12.