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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nA *filter* of $P$ is a subset $F \\subseteq P$ such that:\n\n1. $F$ is nonempty;\n2. if $x \\in F$, $y \\in P$ and $x \\le y$, then $y \\in F$;\n3. if $x, y \\in F$, then there is $z \\in F$ with $z \\le x$ and $z \\le y$.\n\nThe filter $F$ is *proper* if $F \\ne P$.\n\n## Remarks\n\nClause 2 says that $F$ is an [[upper set|ETCE287E2R]].\nIf $P$ is a lattice, clause 3 can be replaced by: $x, y \\in F$ implies $x \\wedge y \\in F$, using the [[meet|J2T75ATRV5]].\nFor $a \\in P$, the set $\\uparrow a = \\set{x \\in P : a \\le x}$ is a filter, the *principal filter* of $a$.\nExample: in the power set of $\\N$, the cofinite sets $\\set{A \\subseteq \\N : \\N \\setminus A \\text{ is finite}}$ form a proper filter.\nIt contains $\\N$, a superset of a cofinite set is cofinite, and $A \\cap B$ is cofinite when $A$ and $B$ are, since $\\N \\setminus (A \\cap B) = (\\N \\setminus A) \\cup (\\N \\setminus B)$.\nNon-example: in the power set of $\\set{1, 2}$, the set $\\set{\\set{1}, \\set{2}, \\set{1, 2}}$ satisfies clauses 1 and 2 but not clause 3.\nThe only common lower bound of $\\set{1}$ and $\\set{2}$ is $\\emptyset$, which is not in the set.\nA filter on a set $X$ in the sense of [[set filters|15PF73BKY6]] is a proper filter of the power set of $X$.\n",
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