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  "body": "\n## Summary\n\nAn orthocomplemented lattice, or ortholattice, is a [[complemented lattice|ZT47BK9Y6K]] with a chosen complement $a^\\perp$ for each element $a$.\nThe map $a \\mapsto a^\\perp$ must reverse the order and be its own inverse.\nThe main examples are [[Boolean algebras|0XADJKM699]], with the Boolean complement.\nAnother is the lattice of closed subspaces of a [[Hilbert space|HM2HHE3J51]], with the [[orthogonal complement|0XJXTBCB2T]].\nThe De Morgan laws hold in every ortholattice.\nOrtholattices are the starting point of the lattice-theoretic approach to [[quantum logic|2E9DMB7A6Q]].\n\n## Explanations\n\n### L1 human\n\nIn a general complemented lattice an element can have many complements, with no preferred one.\nAn orthocomplementation picks one complement for each element in a coherent way, like taking the orthogonal complement of a subspace.\nTaking it twice gives back the element, and a bigger element gets a smaller complement.\nFor a line $\\ell$ through the origin in $\\R^3$, the chosen complement is the plane perpendicular to $\\ell$.\n",
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