Reducible von Neumann Geometries: In Memory of Maurice Audin

Abstract

1. Content of this paper. Given any irreducible von Neumann geometry L and any complete Boolean algebra B we shall construct a particular reducible geometry B(L) which has its centre isomorphic to B and is such that each Iwamura local component of B(L) contains a sublattice isomorphic to L. If B is finite or if L is compact (in the dimension topology) each Iwamura local component is actually isomorphic to L. We shall use a point-free generalization of the constructon given in [2, ?6]. A von Neumann geometry (briefly, a geometry) L is a complemented modular lattice containing at least two elements which is complete and satisfies von Neumann's lattice continuity conditions [6, pp. 1, 2, Axioms I-IV]. Our construction of B(L) is valid and yields a von Neumann geometry whenever (i) B is a Boolean algebra, (ii) L is a complemented modular lattice, and (iii) on L is given a real valued function D(a) such that 0 < D(a) < 1 for each a E L, D(O) = 0, D(1) = 1, and D(a U b) + D(a n b) = D(a)+D(b) for all a, b E L. Thus B(L) should be denoted more precisely as B(L,D). However, as von Neumann showed [6, Part I, Chapter VII], if Lis a von Neumann geometry and irreducible (as defined in [6, p. 3, Axiom VI]) such a dimension function exists and is unique.