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  "body": "\n## Definition\n\nLet $L$ be a [[bounded lattice|72V1Z82M5F]] with least element $0$, greatest element $1$, and [[join and meet|J2T75ATRV5]] $\\vee$ and $\\wedge$.\nA *complement* of an element $a \\in L$ is an element $b \\in L$ such that\n$$a \\wedge b = 0 \\quad\\text{and}\\quad a \\vee b = 1.$$\nThe lattice $L$ is *complemented* if every element of $L$ has at least one complement.\n\n## Remarks\n\nExample: the lattice of subspaces of a finite-dimensional vector space $V$.\nFor a subspace $U$, extend a basis of $U$ to a basis of $V$ and let $W$ be the span of the added vectors; then $U \\cap W = \\set{0}$ and $U + W = V$.\n\nNon-example: the chain $0 \u003c a \u003c 1$.\nThe element $a$ has no complement: $a \\wedge x = 0$ forces $x = 0$, and then $a \\vee x = a \\ne 1$.\n\nIn the [[diamond lattice M3|JQ6FJH2TP8]] each of the three middle elements has the other two as complements, so complements are not unique.\nIn a [[distributive lattice|0HM0HQCK6N]] an element has at most one complement, and a complemented distributive lattice is a [[Boolean algebra|0XADJKM699]].\n",
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