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  "body": "\n## Definition\n\nLet $B$ be a [[Boolean algebra|0XADJKM699]] with order $\\le$.\nA *filter* of $B$ is a subset $F \\subseteq B$ such that $1 \\in F$, $a \\wedge b \\in F$ for all $a, b \\in F$, and $b \\in F$ whenever $a \\in F$ and $a \\le b$.\nIt is *proper* if $0 \\notin F$.\nAn *ultrafilter* of $B$ is a proper filter $U$ such that for every proper filter $F$ with $U \\subseteq F$, one has $F = U$.\n\n## Remarks\n\nThe filters of $B$ are the nonempty [[order filters|Z15FXM9WM1]] of $B$ seen as a lattice.\nFor a proper filter $U$, these are equivalent.\n(a) $U$ is an ultrafilter.\n(b) For every $a \\in B$, exactly one of $a$ and $\\neg a$ lies in $U$.\n(c) $U$ is prime: $a \\vee b \\in U$ implies $a \\in U$ or $b \\in U$.\nUltrafilters $U$ correspond to homomorphisms $\\chi \\colon B \\to \\set{0, 1}$, by $U = \\chi^{-1}(1)$.\n\nExample: in the power set of a set $X$, for $x \\in X$, the family $\\set{A \\subseteq X : x \\in A}$ is an ultrafilter.\nIt is a proper filter, and for each $A$, either $x \\in A$ or $x \\in X \\setminus A$.\n\nNon-example: in the power set of $\\set{1, 2}$, the family $F = \\set{\\set{1, 2}}$ is a proper filter but not an ultrafilter.\nIt contains neither $\\set{1}$ nor its complement $\\set{2}$, and it lies in the larger proper filter $\\set{\\set{1}, \\set{1, 2}}$.\n",
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