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  "body": "\n## Summary\n\nAn ideal of a [[lattice|WH4SFAACAS]] is a nonempty subset that is closed downward and closed under joins.\nThe typical example is the principal ideal $\\downarrow a = \\set{x : x \\le a}$ of all elements below a fixed element $a$.\nLattice ideals are the order-theoretic analogue of [[ideals|096Z9MP2ME]] of a ring, and the dual notion is the [[filter|Z15FXM9WM1]].\nPrime ideals of lattices are the key to representation theorems for distributive lattices and Boolean algebras.\n\n## Explanations\n\n### L1 human\n\nAn ideal of a lattice is a collection of \"small\" elements.\nAnything below a small element is small, and the join of two small elements is still small.\nIn the subsets of the natural numbers, the finite subsets form an ideal: a subset of a finite set is finite, and the union of two finite sets is finite.\nIn the positive integers ordered by divisibility, the divisors of $12$ form an ideal.\n",
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