By William Martin Baker

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Pk) to Ej (resp. P3). I t thus induces an isomorphism of Dfc onto D ; , which we denote by (21) d3k:Dk^D3, 6jk = n(P3k; mk,m3). We define d33 to be the identity. Then the "compatibility" of the 6jk may be formulated as the following lemma. 10. Ifj, (22) k,reJ, then &rj ° Ojk = 6rk- 34 GEOMETRY OF QUANTUM THEORY Proof. From (21) it is immediate t h a t 0kj = 6jk~ . Suppose now t h a t j , Jc, r are three distinct indices from J, XeBk, Y = Ojk(X), Z = 6rj(Y). W e must prove t h a t Z = 6rk(X). W e m a y assume X^O, X^Ek.

As a final reference on the foundational aspects of projective geometry we mention the first volume of H. F . Baker's great six-volume classical treatise ( H . F . B a k e r , Principles of Geometry; originally published b y t h e Cambridge University Press, and republished in 1968 by Frederick Ungar Publishing Co, New York). 4. To von N e u m a n n we owe the most profound and far reaching of the coordinatization theorems for complemented lattices. ). Before doing this the classical theorem of coordinatization h a d to be reformulated so t h a t it made no reference to points; von N e u m a n n viewed it as the establishing of an isomorphism of t h e geometry with the lattice of right ideals of the ring of N x N matrices over some division ring D.

The division rings D^ are of course mutually isomorphic b u t we can do better and introduce a " c o m p a t i b l e " system of isomorphisms. Let j , keJ, j ^ h. The lines E3vEk and P3\/Pk, being in t h e plane m3\imk, meet at some point, say P3k; this is the point at infinity of EjVE^ and the perspectivity from P3k, of the line mk onto the line m3, fixes 0 and takes Ek (resp. Pk) to Ej (resp. P3). I t thus induces an isomorphism of Dfc onto D ; , which we denote by (21) d3k:Dk^D3, 6jk = n(P3k; mk,m3).