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  "body": "\n## Definition\n\nLet $(L, \\vee, \\wedge)$ be a [[lattice|WH4SFAACAS]].\nAn element $a \\in L$ is *join-irreducible* if:\n\n1. $a$ is not a [[least element|0AANKKNHS4]] of $L$;\n2. for all $b, c \\in L$, if $a = b \\vee c$, then $a = b$ or $a = c$.\n\nAn element is *meet-irreducible* if it satisfies the dual conditions, with the greatest element and the [[meet|J2T75ATRV5]].\n\n## Remarks\n\nClause 2 says that $a$ is not the join of two elements strictly below it.\nExample: in the power set of a set $X$, each singleton $\\set{x}$ is join-irreducible.\nIt is not $\\emptyset$, and if $\\set{x} = B \\cup C$ then $B$ or $C$ contains $x$, so it equals $\\set{x}$.\nNon-example: $\\set{1, 2} = \\set{1} \\cup \\set{2}$ with $\\set{1} \\ne \\set{1, 2}$ and $\\set{2} \\ne \\set{1, 2}$, so clause 2 fails.\nIn a chain, every element other than the least element is join-irreducible, since $b \\vee c = \\max(b, c)$.\nIn a finite lattice, every element is the join of the join-irreducible elements below it.\n",
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