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  "body": "\n## Summary\n\nA lattice is a [[partially ordered set|0QV2JS9RJ3]] in which every two elements have a [[join and a meet|J2T75ATRV5]].\nThese are a least upper bound and a greatest lower bound.\nEquivalently, it is a set with two binary operations $\\vee$ and $\\wedge$ that are commutative and associative and satisfy the absorption laws.\nLattices occur all over mathematics.\nSubsets of a set, subgroups of a group, subspaces of a vector space and open sets of a space all form lattices.\nSpecial classes such as [[distributive lattices|0HM0HQCK6N]] and [[Boolean algebras|0XADJKM699]] are central in algebra and logic.\n\n## Explanations\n\n### L1 human\n\nA lattice is an ordering in which any two elements have a best common upper bound and a best common lower bound.\nFor the subsets of a set ordered by inclusion, these are the union and the intersection.\nFor positive integers ordered by divisibility, they are the least common multiple and the greatest common divisor.\nIn a picture of the order, from any two points one can go up to a lowest common point and down to a highest common point.\nNot every ordering is a lattice: two incomparable elements with two incomparable common upper bounds have no best one.\n",
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