Examples of Continuous Geometries.

Construction of the Examples. 1. In the preceding note a system of geometrical axioms was formulated, which is satisfied (i) by the system L = Ln of all linear subspaces of any n1-dimensional projective geometry Pn-,, n = 1, 2, .. (ii) by certain further systems L = L W For each system L satisfying these axioms a unique numerical dimension function D(a) (cf. Definition (12) in sec. 8, loc. cit.) exists, its range being 1 2 (i) D = DX the set (O,, -,) .. ., l) if L = Lni n = lp 2, ....p n n (ii) D = D 0, . 1 if L = L,,. The object of the present note is to give effective examples of the new cases L = LX, and to discuss some of their properties. This note, too, will merely give results and outlines of proofs, the details being reserved for the subsequent publication mentioned in the preceding note. 2. Let ,3 be a not-necessarily-commutative but associative divisionalgebra, and n = 1, 2, . By a left-ratio we mean n elements of 3,