Let $P$ be a finite poset. Let $\zeta:P\times P\to\mathbb C$ be the zeta function of $P$, defined by $\zeta(x,y)=1$ when $x\leq y$ and $\zeta(x,y)=0$ otherwise. Let $\mu:P\times P\to\mathbb C$ be the Möbius function of $P$, meaning the convolution inverse of $\zeta$ in the incidence algebra of $P$. Thus, for all $x,y\in P$ with $x\leq y$,

\begin{align*} \sum_{x\leq z\leq y}\mu(x,z)=\mathbb{1}_{\{x=y\}} \end{align*}

and

\begin{align*} \sum_{x\leq z\leq y}\mu(z,y)=\mathbb{1}_{\{x=y\}}. \end{align*}

If $F,G:P\to\mathbb C$ satisfy

\begin{align*} G(y)=\sum_{x\leq y}F(x) \end{align*}

for every $y\in P$, then

\begin{align*} F(y)=\sum_{x\leq y}G(x)\mu(x,y) \end{align*}

for every $y\in P$.