On continuous geometries. II (Q5972213)

Let \(L\) be a continuous geometry, in general reducible. A mapping of \(L\) into a conditionally complete lattice-group \(G\colon \delta = \delta(x)\) is called a dimension function if each \(\delta \ge 0\) and (1) \(\delta(x + y) = \delta(x) + \delta(y)\) if \(x, y\) are independent; (2) \(\delta\)-equalities are unrestrictedly additive, \(\delta(V^\perp. x_\gamma) = \delta(V^\perp y_\gamma)\) if \(\delta(x_\gamma) = \delta(y_\gamma)\) for each \(\gamma\) \((^\perp\) indicates independent set of elements); (3) \(\delta(a) < \delta(b)\) implies the existence of an \(x <b\) with \(\delta(x) = \delta(a)\); (4) \(0 < f\in G\) implies \(0 < \delta(x) \le f\) for some \(x\in L\); (5) \(\delta(x) = 0\) only if \(x = 0\). Then each such \(\delta\) is necessarily unrestrictedly additive, that is, \(\delta(V^\perp. x_\gamma) = \sum \delta(x_\gamma)\). A special such dimension \(\delta_0(x)\) was constructed in part I [Jap. J. Math. 19, 57--71 (1944; Zbl 0060.32401)] with the additional property: \(\delta_0(x)=\delta_0(y)\) if and only if \(x, y\) are perspective. Thus the author's previous proof that perspectivity is unrestrictedly additive is simplified. Unlike the reviewer's direct proof [Duke Math. J. 5, 503--511 (1939; Zbl 0022.06902)] it still requires the von Neumann dimension theory and his analysis of the centre. The author then considers a congruence relation \(x\sim y\) which includes perspectivity \((x, y\) perspective implies \(x\sim y)\) and for which: \((V^\perp x_\gamma)\sim (V^\perp y_\gamma)\) if \(x_\gamma\sim y_\gamma)\) for each \(\gamma\); \(x\sim( V^\perp y_\gamma)\) implies a decomposition \(x = V^\perp x_\gamma\) with \(x_\gamma\sim y_\gamma\); and \(x\sim x_1\le x\) implies \(x = x_1\). Every dimension function \(\delta(x)\) defines such a relation by: \(x\sim y\) if \(\delta(x) = \delta(y)\). Conversely, every such relation can be so obtained from a dimension function. This is shown by generalizing the results of part I (construction of the dimension function and decomposition of the reducible geometry into a subdirect sum of irreducible components) to the present situation with \(\sim\) in place of perspectivity.