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  "body": "\n## Summary\n\nA complemented lattice is a [[bounded lattice|72V1Z82M5F]] in which every element $a$ has a complement: an element $b$ with $a \\wedge b = 0$ and $a \\vee b = 1$.\nThe lattice of subspaces of a vector space and every [[Boolean algebra|0XADJKM699]] are complemented.\nComplements need not be unique; in a distributive lattice they are, and a complemented distributive lattice is the same as a Boolean algebra.\nComplemented modular lattices are the setting of projective geometry and of von Neumann's [[continuous geometry|5CET1KWWWH]].\n\n## Explanations\n\n### L1 human\n\nA complement of $a$ is something that has nothing in common with $a$ but fills up everything together with $a$.\nFor a line through the origin in the plane, any other line through the origin is a complement.\nSo a complement exists, but there are many choices.\nIn the lattice of subsets of a set, the only complement of a subset is the set of points outside it.\n",
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