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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]], and write $x \u003c y$ for $x \\le y$ and $x \\ne y$.\nFor $x, y \\in P$, the element $y$ *covers* $x$, written $x \\lessdot y$, when both conditions hold.\n\n1. $x \u003c y$.\n2. There is no $z \\in P$ with $x \u003c z \u003c y$.\n\nThe covering relation of $P$ is the set of pairs $(x, y)$ with $x \\lessdot y$.\n\n## Remarks\n\nOrder the positive integers by divisibility.\nThen $2 \\lessdot 6$: the only divisors of $6$ that $2$ divides are $2$ and $6$, so nothing lies strictly between.\nIn general $m \\lessdot n$ exactly when $m \\mid n$ and $n / m$ is prime.\nThe pair $2 \u003c 8$ is not a cover, since $2 \u003c 4 \u003c 8$, so the second condition fails.\nIn a finite poset, $x \u003c y$ holds if and only if there is a finite sequence $x = z_0 \\lessdot z_1 \\lessdot \\cdots \\lessdot z_k = y$ with $k \\ge 1$.\nSo the covering relation determines a finite partial order.\nThis fails for infinite posets: in $\\Q$ the covering relation is empty.\n\n## Explanations\n\n### L1 human\n\nA cover is a single step up in the order, with no room for anything in between.\nThe lines of a Hasse diagram are exactly the covers.\n",
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