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  "body": "\n## Summary\n\nA semimodular lattice is a [[lattice|WH4SFAACAS]] in which joining with an element preserves the covering relation: if $a$ covers $a \\wedge b$, then $a \\vee b$ covers $b$.\nIt is a one-sided weakening of the [[modular law|QX7WZ92QAK]], and every modular lattice is semimodular.\nLattices of flats of [[matroids|PX31ZNYS3Z]] and lattices of partitions of a set are semimodular without being modular in general.\nIn lattices of finite length, semimodularity is what makes a well-behaved rank function exist.\n\n## Explanations\n\n### L1 human\n\nThink of the elements of a lattice as subspaces and of a cover as a step up by one dimension.\nSemimodularity says: if $a$ is one step above $a \\wedge b$, then adding $a$ to $b$ moves $b$ up by at most one step, and in fact by exactly one.\nSo \"one more generator\" never gives more than \"one more dimension\".\nIn the lattice of partitions of a set, merging two blocks is one step up, and this rule holds there too.\n",
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