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  "body": "\n## Definition\n\nLet $(L, \\vee, \\wedge)$ be a [[lattice|WH4SFAACAS]] with order $\\le$.\nAn element $a \\in L$ is *join-prime* if:\n\n1. $a$ is not a [[least element|0AANKKNHS4]] of $L$;\n2. for all $b, c \\in L$, if $a \\le b \\vee c$, then $a \\le b$ or $a \\le c$.\n\n## Remarks\n\nEvery join-prime element is [[join-irreducible|ZHRQ0AT8PM]]: if $a = b \\vee c$, then $a \\le b$ or $a \\le c$, and since $b, c \\le a$ this gives $a = b$ or $a = c$.\nIn a [[distributive lattice|0HM0HQCK6N]] the converse holds: if $a$ is join-irreducible and $a \\le b \\vee c$, then $a = a \\wedge (b \\vee c) = (a \\wedge b) \\vee (a \\wedge c)$, so $a = a \\wedge b$ or $a = a \\wedge c$.\nExample: in the power set of a set, a singleton $\\set{x}$ is join-prime, since $x \\in B \\cup C$ implies $x \\in B$ or $x \\in C$.\nNon-example: in the diamond lattice $M_3 = \\set{0, a, b, c, 1}$, where $a$, $b$, $c$ are pairwise incomparable and the join of any two of them is $1$, the element $a$ is join-irreducible.\nIt is not join-prime, because $a \\le b \\vee c = 1$ while $a \\not\\le b$ and $a \\not\\le c$.\nMeet-prime elements are defined dually.\n",
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