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  "body": "\n## Definition\n\nLet $B$ be a [[Boolean algebra|0XADJKM699]] with order $\\le$.\nAn [[atom|TTVCTW6TNC]] of $B$ is an element $a \\ne 0$ such that no $b \\in B$ satisfies $0 \u003c b \u003c a$.\nThe algebra $B$ is *atomless* if it has no atoms, that is, for every $a \\ne 0$ in $B$ there is $b \\in B$ with $0 \u003c b \u003c a$.\n\n## Remarks\n\nThe one-element algebra, with $0 = 1$, is atomless in a trivial way.\nEvery other atomless Boolean algebra is infinite, since in a finite algebra with $0 \\ne 1$ a minimal element of $B \\setminus \\set{0}$ is an atom.\n\nExample: let $B$ be the algebra of finite unions of half-open intervals $[s, t)$ with $0 \\le s \u003c t \\le 1$, together with $\\emptyset$, as subsets of $[0, 1)$.\nA nonzero $a \\in B$ contains some $[s, t)$ with $s \u003c t$; then $b = [s, (s + t)/2)$ lies in $B$ and satisfies $\\emptyset \\ne b \\subsetneq a$.\n\nNon-example: the power set of a nonempty set $X$ is not atomless, since each singleton $\\set{x}$ is an atom.\n",
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