common complement. This is an equivalence relation on L; the proof that it is transitive is quite hard. The equivalence classes A, B, ... of L have a total order on them defined by if there is some a in A and b in B with . (This need not hold for all a in A and b in B.) The dimension function D from L to the unit interval is defined as follows. If equivalence classes A and B contain elements a and b with then their sum is defined to be the equivalence class of . Otherwise the sum is not defined. For a positive integer n, the product nA is defined to be the sum of n copies of A, if this sum is defined. For equivalence classes A and B with A not {0} the integer is defined to be the unique integer such that with . For equivalence classes A and B with A not {0} the real number is defined to be the limit of as C runs through a minimal sequence: this means that either C contains a minimal nonzero element, or an infinite sequence of nonzero elements each of which is at most half the preceding one. D(a) is defined to be , where {a} and {1} are the equivalence classes containing a and 1. The image of D can be the whole unit interval, or the set of numbers for some positive integer n. Two elements of L have the same image under D if and only if they are perspective, so it gives an injection from the equivalence classes to a subset of the unit interval. The dimension function D has the properties: If then D(a ∨ b) + D(a ∧ b) = D(a) + D(b) if and only if , and if and only if Coordinatization theorem In projective geometry, the Veblen–Young theorem states that a projective geometry of dimension at least 3 is isomorphic to the projective geometry of a vector space over a division ring. This can be restated as saying that the subspaces in the projective geometry correspond to the principal right ideals of a matrix algebra over a division ring. Neumann generalized this to continuous geometries, and more generally to complemented modular lattices, as follows . His theorem states that if a complemented modular lattice L has order at least 4, then the elements of