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  "body": "\n## Definition\n\nLet $L$ be a [[bounded lattice|72V1Z82M5F]] with least element $0$ and greatest element $1$.\nAn *orthocomplementation* on $L$ is a map $a \\mapsto a^\\perp$ from $L$ to $L$ such that for all $a, b \\in L$:\n\n1. $a \\wedge a^\\perp = 0$ and $a \\vee a^\\perp = 1$, so $a^\\perp$ is a [[complement|ZT47BK9Y6K]] of $a$;\n2. $(a^\\perp)^\\perp = a$;\n3. if $a \\le b$, then $b^\\perp \\le a^\\perp$.\n\nAn *orthocomplemented lattice*, or *ortholattice*, is a bounded lattice together with an orthocomplementation.\n\n## Remarks\n\nFrom clauses 2 and 3 follow the De Morgan laws $(a \\vee b)^\\perp = a^\\perp \\wedge b^\\perp$ and $(a \\wedge b)^\\perp = a^\\perp \\vee b^\\perp$.\n\nExample: the lattice of closed linear subspaces of a [[Hilbert space|HM2HHE3J51]] $H$, with $M^\\perp$ the [[orthogonal complement|0XJXTBCB2T]] of $M$.\nHere $M \\cap M^\\perp = \\set{0}$, the closed span of $M$ and $M^\\perp$ is $H$, $M^{\\perp\\perp} = M$ for closed $M$, and $M \\subseteq N$ implies $N^\\perp \\subseteq M^\\perp$.\n\nNon-example: the [[diamond lattice M3|JQ6FJH2TP8]] with middle elements $a, b, c$ and the map $a \\mapsto b$, $b \\mapsto c$, $c \\mapsto a$, $0 \\mapsto 1$, $1 \\mapsto 0$.\nEach image is a complement, but $(a^\\perp)^\\perp = c \\ne a$, so clause 2 fails.\n",
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