Geodesic
The M\"obius inverse formulas on Abelian semigroups
Čebyševskij sbornik, Tome 10 (2009) no. 2, pp. 55-78.

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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.
Keywords: Locally finite Abelian semigroup, ideal, idempotent, character, $\zeta$–functions, algebraic invertibility. 
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E. A. Gorin. The M\"obius inverse formulas on Abelian semigroups. Čebyševskij sbornik, Tome 10 (2009) no. 2, pp. 55-78. http://geodesic.mathdoc.fr/item/CHEB_2009_10_2_a2/

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