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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nFor $x, y \\in P$, say that $y$ covers $x$ when $x \u003c y$ and no $z \\in P$ satisfies $x \u003c z \u003c y$; this is the [[covering relation|4HHDPZ4BDY]].\nA *rank function* on $P$ is a map $\\rho \\colon P \\to \\Z$ such that:\n\n1. if $x \u003c y$, then $\\rho(x) \u003c \\rho(y)$;\n2. if $y$ covers $x$, then $\\rho(y) = \\rho(x) + 1$.\n\nA *graded poset* is a partially ordered set together with a rank function.\n\n## Remarks\n\nExample: the power set of a finite set, ordered by inclusion, is graded by $\\rho(A) = \\abs{A}$.\nA subset $B$ covers $A$ exactly when $B = A \\cup \\set{b}$ with $b \\notin A$, so the rank goes up by one.\nExample: the positive divisors of an integer $n \\ge 1$, ordered by divisibility, are graded by the number of prime factors counted with multiplicity.\nNon-example: the [[pentagon lattice|DHC8TZXGYS]] with elements $0 \u003c a \u003c b \u003c 1$ and $0 \u003c c \u003c 1$ has no rank function.\nThe covers $0, a, b, 1$ would give $\\rho(1) = \\rho(0) + 3$, while the covers $0, c, 1$ would give $\\rho(1) = \\rho(0) + 2$.\nSome authors also require $\\rho$ to take values in $\\N$, or require all maximal chains to have the same length; the conventions should be checked in each source.\n",
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