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  "body": "\n## Statement\n\nLet $L$ be a finite [[distributive lattice|0HM0HQCK6N]].\nLet $J(L)$ be the set of [[join-irreducible elements|ZHRQ0AT8PM]] of $L$.\nThese are the elements $j$, other than the least element, such that $j = x \\vee y$ implies $j = x$ or $j = y$.\nOrder $J(L)$ by the order of $L$, and let $\\mathcal{O}(J(L))$ be the set of lower sets of the [[partially ordered set|0QV2JS9RJ3]] $J(L)$, ordered by inclusion.\nThen the map\n$$a \\mapsto \\set{j \\in J(L) : j \\le a}$$\nis a lattice isomorphism from $L$ onto $\\mathcal{O}(J(L))$, where meet and join in $\\mathcal{O}(J(L))$ are intersection and union.\n\n## Remarks\n\nEvery element $a$ of a finite lattice is the join of the join-irreducible elements below it, so the map is injective and order-preserving in both directions.\nDistributivity is used for surjectivity: if $j \\le x \\vee y$ with $j$ join-irreducible, then $j = (j \\wedge x) \\vee (j \\wedge y)$, so $j \\le x$ or $j \\le y$.\nThis shows that the map sends joins to unions, and so every lower set $D$ is the image of $\\bigvee D$.\n\nExample: the divisors of $12$ ordered by divisibility form a distributive lattice.\nIts join-irreducible elements are the prime powers $2$, $4$ and $3$, with $2 \\le 4$ and $3$ incomparable to both.\nThis poset has six lower sets, matching the six divisors $1, 2, 3, 4, 6, 12$; for example $6$ corresponds to $\\set{2, 3}$.\n\nDistributivity is necessary: in the [[diamond lattice M3|JQ6FJH2TP8]] all three middle elements are join-irreducible and pairwise incomparable, so the lower sets of $J(M_3)$ number $8$, while $M_3$ has $5$ elements.\n\n## Explanations\n\n### L1 human\n\nEvery finite distributive lattice looks like a family of sets closed under union and intersection.\nThe building blocks are the join-irreducible elements, which cannot be split as a join of smaller elements.\nEach element is recorded by the set of building blocks below it.\n\n### L2 human\n\nThe theorem gives a correspondence between finite posets and finite distributive lattices: a poset $P$ gives the lattice of its lower sets, and a lattice $L$ gives the poset $J(L)$.\nThis correspondence extends to a duality between the category of finite posets with order-preserving maps and the category of finite distributive lattices with bounded lattice homomorphisms.\nFor infinite distributive lattices, the analogue needs topology, as in [[Priestley duality|SA22SEC75K]].\nIn the infinite case, prime filters take the place of join-irreducible elements, and the [[prime ideal theorem for distributive lattices|ME8GM0VTWP]] guarantees that there are enough of them.\n",
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