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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]], and write $a \u003c b$ when $a \\le b$ and $a \\ne b$.\nA *successor order* on $P$ is a map $\\operatorname{succ} \\colon P \\to P$ such that:\n\n1. $a \\le \\operatorname{succ}(a)$ for all $a \\in P$.\n2. If $\\operatorname{succ}(a) \\le a$, then $a$ is a maximal element of $P$.\n3. If $a \u003c b$, then $\\operatorname{succ}(a) \\le b$.\n\n## Remarks\n\nWhen $a$ is not maximal, the clauses say that $\\operatorname{succ}(a)$ is the least element of $\\set{b \\in P : a \u003c b}$; when $a$ is maximal, $\\operatorname{succ}(a) = a$.\nSo the map, if it exists, is unique.\n\nExample: $\\Z$ with $\\operatorname{succ}(n) = n + 1$.\nCheck: $n \\le n + 1$; the inequality $n + 1 \\le n$ never holds, so clause 2 is empty; and $n \u003c m$ for integers means $n + 1 \\le m$.\nOn $\\N$ this is the [[successor function|0AK9TGSNXY]].\n\nNon-example: $\\Q$ with the usual order has no successor order.\nNo rational is maximal, so clauses 1 and 2 force $a \u003c \\operatorname{succ}(a)$; then $b = (a + \\operatorname{succ}(a))/2$ satisfies $a \u003c b \u003c \\operatorname{succ}(a)$, which breaks clause 3.\n\nThe dual notion is a predecessor order.\nIn Mathlib this is the class `SuccOrder`.\n",
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