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  "body": "\n## Definition\n\nLet $L$ be a [[lattice|WH4SFAACAS]] with a [[least element|0AANKKNHS4]] $0$.\n$L$ is *atomic* if for every $b \\in L$ with $b \\ne 0$ there is an [[atom|TTVCTW6TNC]] $a$ of $L$ with $a \\le b$.\n\n## Remarks\n\nThe same definition applies to any partially ordered set with a least element.\nExample: every finite lattice $L$ is atomic.\nFor $b \\ne 0$, the set $\\set{x \\in L : 0 \u003c x \\le b}$ is finite and nonempty, so it has a minimal element, and a minimal element of this set is an atom.\nExample: $\\N_{\u003e0}$ ordered by divisibility is atomic, since every $n \u003e 1$ has a prime divisor.\nNon-example: the chain $[0, 1]$ with its usual order is a lattice with least element $0$ and no atoms, so the condition fails for $b = 1$.\n",
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  "msc": [
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  "name": "Atomic lattice",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/GRM/GRMT65QJAY.md",
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