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  "body": "\n## Definition\n\nA *lattice* is a [[partially ordered set|0QV2JS9RJ3]] $(L, \\le)$ in which every two elements $x, y \\in L$ have a [[join|J2T75ATRV5]] $x \\vee y$ and a meet $x \\wedge y$.\n\nAlgebraic form: a *lattice* is a set $L$ with two binary operations $\\vee$ and $\\wedge$ such that, for all $x, y, z \\in L$:\n\n1. $x \\vee y = y \\vee x$ and $x \\wedge y = y \\wedge x$;\n2. $(x \\vee y) \\vee z = x \\vee (y \\vee z)$ and $(x \\wedge y) \\wedge z = x \\wedge (y \\wedge z)$;\n3. $x \\vee (x \\wedge y) = x$ and $x \\wedge (x \\vee y) = x$.\n\n## Remarks\n\nThe two forms are equivalent.\nFrom a lattice order, the join and meet satisfy clauses 1 to 3.\nFrom the operations, the order is recovered by $x \\le y \\iff x \\vee y = y$, which holds exactly when $x \\wedge y = x$.\nThe idempotent laws $x \\vee x = x$ and $x \\wedge x = x$ follow from the absorption laws in clause 3.\nExample: $\\N_{\u003e0}$ ordered by divisibility is a lattice with $a \\vee b = \\lcm(a, b)$ and $a \\wedge b = \\gcd(a, b)$.\nIndeed $\\lcm(a, b)$ is a common multiple of $a$ and $b$ and divides every common multiple.\nNon-example: the poset $\\set{a, b, c, d}$ with $a, b \u003c c$ and $a, b \u003c d$ and no other strict relations is not a lattice.\nThe elements $a$ and $b$ have the two upper bounds $c$ and $d$, which are incomparable, so $a \\vee b$ does not exist.\n",
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