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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nFor $x \\le y$ in $P$, let $[x, y] = \\set{z \\in P : x \\le z \\le y}$.\nA finite [[chain|0S8FE7ZMFM]] $z_0 \u003c z_1 \u003c \\cdots \u003c z_n$ has length $n$.\nThe poset $P$ satisfies the *Jordan–Dedekind chain condition* when the following holds for all $x \\le y$ in $P$.\nAll [[maximal chains|3AX549FFFM]] of $[x, y]$ are finite and have the same length.\n\n## Remarks\n\nExample: the positive divisors of $12$ under divisibility satisfy the condition.\nA maximal chain of $[a, b]$ multiplies by one prime at each step, so its length is the number of prime factors of $b / a$ counted with multiplicity.\nNon-example: the pentagon with $0 \u003c a \u003c b \u003c 1$ and $0 \u003c c \u003c 1$ fails it, since $0 \u003c a \u003c b \u003c 1$ and $0 \u003c c \u003c 1$ are maximal chains of $[0, 1]$ with lengths $3$ and $2$.\nLet $P$ be finite with a least element $\\hat{0}$.\nThen $P$ satisfies the condition if and only if $P$ has a rank function.\nIn that case, setting $\\rho(x)$ equal to the length of any maximal chain of $[\\hat{0}, x]$ gives a rank function.\nEvery modular lattice of finite length satisfies the condition.\n",
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  "name": "Jordan–Dedekind chain condition",
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