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  "body": "\n## Definition\n\nLet $L$ be a [[lattice|WH4SFAACAS]] with a [[least element|0AANKKNHS4]] $0$.\n$L$ is *atomistic* if every $b \\in L$ is the [[join|J2T75ATRV5]] in $L$ of the set $\\set{a \\in L : a \\text{ is an atom and } a \\le b}$.\n\n## Remarks\n\nThe least element $0$ is the join of the empty set of atoms.\nAn atomistic lattice is [[atomic|5SE8EMSDNY]]: if $b \\ne 0$ had no atom below it, $b$ would be the join of the empty set, that is $b = 0$.\nExample: the power set of a set $X$ is atomistic, since every subset $B$ is the union of the singletons $\\set{x}$ with $x \\in B$.\nNon-example: the chain $0 \u003c 1 \u003c 2$ is atomic, with the single atom $1$, but not atomistic.\nThe join of the atoms below $2$ is $1 \\ne 2$.\n",
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