Geodesic
The M\"obius Function on Abelian Semigroups
Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 88-93.

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Let $X$ be an Abelian semigroup such that the following conditions hold: (i) if $x\times y=1\mspace{-4.85mu}{\mathrm I}$ ($1\mspace{-4.85mu}{\mathrm I}$ is the identity element), then $x=y=1\mspace{-4.85mu}{\mathrm I}$; (ii) the set $\{\{x,y\}\colon x\times y=a\}$ is finite for any $a\in X$. Let $\Lambda$ be any field, and let $\mathcal{E}$ be the algebra of all $\Lambda$-valued functions on $X$. The convolution of $u,v\in\mathcal{E}$ is defined by $$ (u*v)(x)=\sum\{u(a)v(b)\colon a\times b=x\}. $$ We set $\varepsilon(x)=1_{\Lambda}$ for all $x\in X$. The Möbius function $\mu$ is defined as the solution of the equation $\varepsilon*\mu=\delta$ ($\delta$ is the Dirac function). The Möbius function is unique (if it exists at all). Some existence conditions are given. If $\Lambda$ is replaced by the ring of integers, then $\mu$ exists if and only if $X$ does not contain nontrivial idempotents.
Keywords: Abelian semigroup, free module, $\zeta$-functions. 
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E. A. Gorin. The M\"obius Function on Abelian Semigroups. Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 88-93. http://geodesic.mathdoc.fr/item/FAA_2011_45_1_a8/

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