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  "body": "\n## Summary\n\nA filter of a [[partially ordered set|0QV2JS9RJ3]] is a nonempty subset that is closed upward.\nMoreover, any two of its elements have a common lower bound inside it.\nIn a [[lattice|WH4SFAACAS]] this means: closed upward and closed under meets.\nFilters are the order duals of [[lattice ideals|5GW2T2AM1A]].\nA [[filter on a set|15PF73BKY6]] $X$ is a proper order filter in the lattice of subsets of $X$.\n\n## Explanations\n\n### L1 human\n\nAn order filter is a collection of \"large\" elements.\nAnything above a large element is large, and two large elements always have a large element below both of them.\nFor a fixed element $a$, the set of all elements above $a$ is a filter.\nIn the subsets of the natural numbers, the sets with finite complement form a filter, since the intersection of two such sets again has finite complement.\n",
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