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  "body": "\n## Summary\n\nAn element $a$ of a [[lattice|WH4SFAACAS]] is join-prime if it is not the bottom element and has the following property.\nWhenever $a$ lies below a join $b \\vee c$, it lies below $b$ or below $c$.\nThe definition copies that of a [[prime element|01VH21EV6F]] of a ring, with \"divides\" replaced by \"lies below\" and the product replaced by the [[join|J2T75ATRV5]].\nEvery join-prime element is [[join-irreducible|ZHRQ0AT8PM]], and in a [[distributive lattice|0HM0HQCK6N]] the two notions coincide.\n\n## Explanations\n\n### L1 human\n\nA join-prime element cannot hide inside a join without hiding inside one of the two parts.\nIn the subsets of a set, a singleton $\\set{x}$ is join-prime: if $x$ is in $B \\cup C$, then $x$ is in $B$ or in $C$.\nIn the lattice $\\set{0, a, b, c, 1}$ with three incomparable middle elements, $a$ is join-irreducible but not join-prime.\nIt lies below $b \\vee c = 1$ but not below $b$ or $c$.\n",
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