Möbius function of a poset

Summary

The Möbius function of a locally finite poset assigns an integer $\mu(x, y)$ to each pair $x \le y$, by the recursion $\mu(x, x) = 1$ and $\sum_{x \le z \le y} \mu(x, z) = 0$ for $x < y$. It is the inverse of the zeta function in the incidence algebra. It drives Möbius inversion on posets, which contains inclusion–exclusion and the number-theoretic Möbius function as special cases.