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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]] with a [[least element|0AANKKNHS4]] $0$.\nAn *atom* of $P$ is an element $a \\in P$ such that:\n\n1. $0 \u003c a$;\n2. there is no $x \\in P$ with $0 \u003c x \u003c a$.\n\n## Remarks\n\nEquivalently, $a$ is an atom when $a$ [[covers|4HHDPZ4BDY]] $0$, or when $a$ is a minimal element of $P \\setminus \\set{0}$.\nExample: in $\\N_{\u003e0}$ ordered by divisibility, with least element $1$, every prime $p$ is an atom.\nIndeed $1 \\ne p$, and the only divisors of $p$ are $1$ and $p$, so nothing lies strictly between them.\nNon-example: $4$ is not an atom of this poset, since $1 \u003c 2 \u003c 4$ in the divisibility order, so clause 2 fails.\nThe interval $[0, 1]$ with its usual order has no atoms, because $0 \u003c a / 2 \u003c a$ for each $a \u003e 0$.\nThe dual notion, an element just below a greatest element, is a *coatom*.\n",
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