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  "body": "\n## Summary\n\nA prime ideal of a [[lattice|WH4SFAACAS]] is a proper [[ideal|5GW2T2AM1A]] that contains one of $x$ or $y$ whenever it contains their meet $x \\wedge y$.\nIt is the lattice analogue of a [[prime ideal|00MHHZ9SNE]] of a commutative ring, with the meet in the role of the product.\nThe complement of a prime ideal is a prime [[filter|Z15FXM9WM1]].\nPrime ideals are the points of the spaces used to represent distributive lattices and Boolean algebras.\n\n## Explanations\n\n### L1 human\n\nAn ideal is prime when it cannot catch the common part of two elements without catching one of the two elements.\nFix a point $p$ of a set $X$ and look at all subsets of $X$ that do not contain $p$.\nThey form a prime ideal of the subsets of $X$: if $A \\cap B$ misses $p$, then $A$ or $B$ misses $p$.\nSplitting the lattice into this ideal and its complement is like asking the single yes-or-no question \"does the set contain $p$?\".\n",
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