The integer-valued Möbius function $\mu_P$ of a locally finite poset

Let $P$ be a locally finite poset. Take coefficients in the commutative ring $\mathbb Z$ (The integers form a commutative ring). The zeta incidence function has diagonal value $1$, hence is convolution-invertible by An incidence function is convolution-invertible if and only if every diagonal value is a unit. The Möbius function of $P$ is its unique inverse

$\mu_P:=\zeta^{-1}\in I(P,\mathbb Z),$

so

$\mu_P*\zeta=\delta=\zeta*\mu_P$

with $\delta$ and $\zeta$ as in The delta and zeta incidence functions. Its value $\mu_P(x,y)$ is therefore an integer for every $x\le y$.

The coefficient ring is fixed as $\mathbb Z$. When a formula takes values in another ring $R$, the integer $\mu_P(x,y)$ acts through its canonical repeated-addition multiple of $1_R$; no characteristic-dependent second Möbius function is introduced.