Continuous geometry

Remarks

Some authors omit the irreducibility clause 5 and call the irreducible ones continuous geometries in the strict sense.

Example: the lattice of subspaces of $D^n$ for a division ring $D$ and $n \ge 2$. It is modular and complemented; it is complete and clause 4 holds because every chain in it is finite. It is irreducible: a subspace $U \ne 0, D^n$ has more than one complement. Indeed, a complement $W$ can be changed by adding a nonzero vector of $U$ to one basis vector of $W$.

Non-example: the lattice of all subsets of a set $X$ with at least two elements. It satisfies clauses 1 to 4, but every subset has exactly one complement, so clause 5 fails.

Von Neumann proved that a continuous geometry has a dimension function $d \colon L \to [0, 1]$ with $d(0) = 0$, $d(1) = 1$ and $d(a \vee b) + d(a \wedge b) = d(a) + d(b)$. Moreover $d(a) = d(b)$ if and only if $a$ and $b$ have a common complement. Its range is either $\set{0, 1/n, \dots, 1}$ for some $n \ge 1$ or the whole interval $[0, 1]$.