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  "body": "\n## Definition\n\nLet $L$ be a finite [[lattice|WH4SFAACAS]].\nThe lattice $L$ is *geometric* if:\n\n1. $L$ is [[atomistic|99DVVV7BH2]]: every element of $L$ is the join of a set of [[atoms|TTVCTW6TNC]];\n2. $L$ is [[semimodular|C1NET23GG0]] (upper semimodular).\n\n## Remarks\n\nExample: the Boolean lattice of all subsets of a finite set $S$.\nThe atoms are the singletons, and every subset is the union of its singletons.\nIf $A$ covers $A \\cap B$, then $A = (A \\cap B) \\cup \\set{s}$ with $s \\notin B$, so $A \\cup B = B \\cup \\set{s}$ covers $B$.\n\nNon-example: the chain $0 \u003c a \u003c 1$ with three elements.\nIt is semimodular, but its only atom is $a$, and $1$ is not a join of atoms.\n\nThe lattice of flats of a finite [[matroid|PX31ZNYS3Z]] is geometric, and every geometric lattice is isomorphic to the lattice of flats of a simple matroid, unique up to isomorphism.\nSome authors allow infinite geometric lattices, replacing finiteness by finite length or by compactness conditions.\n",
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