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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[locally finite poset|TV1V4JDRDD]], and write $x \u003c y$ when $x \\le y$ and $x \\ne y$.\nThe *Möbius function* of $P$ is the function $\\mu \\colon P \\times P \\to \\Z$ given by:\n\n1. $\\mu(x, x) = 1$ for all $x \\in P$.\n2. $\\mu(x, y) = -\\sum_{x \\le z \u003c y} \\mu(x, z)$ for $x \u003c y$.\n3. $\\mu(x, y) = 0$ when $x \\not\\le y$.\n\n## Remarks\n\nClause 2 is a recursion over the finite interval $[x, y]$, so $\\mu$ is well defined.\nEquivalently, $\\mu$ is the inverse of the zeta function $\\zeta$ in the [[incidence algebra|1H9SFWHYEB]] of $P$, where $\\zeta(x, y) = 1$ for $x \\le y$ and $0$ otherwise.\nFrom this one also gets $\\mu(x, y) = -\\sum_{x \u003c z \\le y} \\mu(z, y)$ for $x \u003c y$.\n\nExample: in the lattice of subsets of a finite set, $\\mu(S, T) = (-1)^{\\abs{T \\setminus S}}$ for $S \\subseteq T$.\nCheck: for $\\abs{T \\setminus S} = k \\ge 1$, the sum $\\sum_{S \\subseteq U \\subseteq T} (-1)^{\\abs{U \\setminus S}} = \\sum_{j=0}^{k} \\binom{k}{j} (-1)^j = (1 - 1)^k = 0$, as clauses 1 and 2 require.\n\nNon-example: the function $\\nu(x, y) = (-1)^{y - x}$ for $x \\le y$ on the chain $\\N$ is not the Möbius function.\nIt satisfies clause 1, but clause 2 fails at $(0, 2)$: there $\\mu(0, 2) = -(1 - 1) = 0$, while $\\nu(0, 2) = 1$.\n\nFor the positive integers ordered by divisibility, $\\mu(d, n)$ is the number-theoretic Möbius function of $n / d$.\n",
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