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  "body": "\n## Example\n\nLet $\\Z_{\u003e0}$ be the set of positive integers, and write $a \\mid b$ when $a$ [[divides|0PG2H7WKGG]] $b$.\nThe relation $\\mid$ is a [[partial order|0QV2JS9RJ3]] on $\\Z_{\u003e0}$.\nIt is reflexive since $a = 1 \\cdot a$.\nIt is antisymmetric since $a \\mid b$ and $b \\mid a$ give $a \\le b \\le a$ for positive integers.\nIt is transitive since $b = ka$ and $c = lb$ give $c = (kl) a$.\n\nIt is a [[lattice|WH4SFAACAS]]: the meet of $a$ and $b$ is their [[greatest common divisor|0C8WYW4KYQ]], and the join is their [[least common multiple|007NPQX34K]].\nIt is a [[distributive lattice|0HM0HQCK6N]].\nBy the [[fundamental theorem of arithmetic|084HH1NDCB]], write $a$, $b$, $c$ as products of primes.\nFor each prime $p$, gcd takes the minimum and lcm takes the maximum of the exponents of $p$.\nThe identity $\\min(x, \\max(y, z)) = \\max(\\min(x, y), \\min(x, z))$ for integers then gives $\\gcd(a, \\operatorname{lcm}(b, c)) = \\operatorname{lcm}(\\gcd(a, b), \\gcd(a, c))$.\n\n## Remarks\n\nThe element $1$ is the least element, and there is no greatest element in $\\Z_{\u003e0}$.\nIf $0$ is added, it becomes the greatest element, since every integer divides $0$.\nThe positive divisors of a fixed $n$ form a finite sublattice; for $n = p_1^{e_1} \\cdots p_k^{e_k}$ it is isomorphic to a product of chains with $e_1 + 1, \\ldots, e_k + 1$ elements.\nThe order is not total: $2$ and $3$ are incomparable.\n\n## Explanations\n\n### L1 human\n\nOrder the positive integers by \"divides\" instead of \"is smaller than\".\nThen $2$ is below $6$, but $2$ and $3$ are side by side, and the primes sit just above $1$.\nThe greatest common divisor and the least common multiple play the roles of \"meet\" and \"join\".\n",
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