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  "body": "\n## Definition\n\nLet $L$ be a lattice with join $\\vee$ and meet $\\wedge$.\nThe lattice $L$ is a *continuous geometry* if:\n\n1. $L$ is a [[modular lattice|QX7WZ92QAK]];\n2. $L$ is a [[complete lattice|D7A0ZGZT5S]], with least element $0$ and greatest element $1$;\n3. $L$ is [[complemented|ZT47BK9Y6K]];\n4. for every [[chain|0S8FE7ZMFM]] $C \\subseteq L$ and every $b \\in L$,\n$$b \\wedge \\bigvee_{c \\in C} c = \\bigvee_{c \\in C} (b \\wedge c) \\quad\\text{and}\\quad b \\vee \\bigwedge_{c \\in C} c = \\bigwedge_{c \\in C} (b \\vee c);$$\n5. $L$ is irreducible: the only elements of $L$ that have exactly one complement are $0$ and $1$.\n\n## Remarks\n\nSome authors omit the irreducibility clause 5 and call the irreducible ones continuous geometries in the strict sense.\n\nExample: the lattice of subspaces of $D^n$ for a division ring $D$ and $n \\ge 2$.\nIt is modular and complemented; it is complete and clause 4 holds because every chain in it is finite.\nIt is irreducible: a subspace $U \\ne 0, D^n$ has more than one complement.\nIndeed, a complement $W$ can be changed by adding a nonzero vector of $U$ to one basis vector of $W$.\n\nNon-example: the lattice of all subsets of a set $X$ with at least two elements.\nIt satisfies clauses 1 to 4, but every subset has exactly one complement, so clause 5 fails.\n\nVon Neumann proved that a continuous geometry has a dimension function $d \\colon L \\to [0, 1]$ with $d(0) = 0$, $d(1) = 1$ and $d(a \\vee b) + d(a \\wedge b) = d(a) + d(b)$.\nMoreover $d(a) = d(b)$ if and only if $a$ and $b$ have a common complement.\nIts range is either $\\set{0, 1/n, \\dots, 1}$ for some $n \\ge 1$ or the whole interval $[0, 1]$.\n",
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