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  "body": "\n## Definition\n\nLet $(L, \\vee, \\wedge)$ be a [[lattice|WH4SFAACAS]] with its order $\\le$.\nAn *ideal* of $L$ is a subset $I \\subseteq L$ such that:\n\n1. $I$ is nonempty;\n2. if $x \\in I$, $y \\in L$ and $y \\le x$, then $y \\in I$;\n3. if $x, y \\in I$, then $x \\vee y \\in I$.\n\nThe ideal $I$ is *proper* if $I \\ne L$.\n\n## Remarks\n\nFor $a \\in L$, the set $\\downarrow a = \\set{x \\in L : x \\le a}$ is an ideal, the *principal ideal* generated by $a$.\nIn a finite lattice every ideal is principal, generated by the join of all its elements.\nExample: in the power set of $\\N$ ordered by $\\subseteq$, the finite subsets form an ideal.\nIt is nonempty, every subset of a finite set is finite, and the union of two finite sets is finite.\nIt is not principal, since no finite set contains all finite sets.\nNon-example: in the power set of $\\set{1, 2}$, the set $\\set{\\emptyset, \\set{1}, \\set{2}}$ satisfies clauses 1 and 2 but not clause 3, since $\\set{1} \\cup \\set{2} = \\set{1, 2}$ is missing.\nThe same notion makes sense in any [[partially ordered set|0QV2JS9RJ3]], with clause 3 replaced by: any two elements of $I$ have an upper bound in $I$.\nIn a lattice the two versions agree.\n",
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