Let L be a complemented, χ-complete modular lattice. A theorem of Amemiya
and Halperin (see [l], Theorem 4.3) asserts that if the intervals [O, a] and
[O, b], a, bεL, are upper χ-continuous then [O, a∪b] is also upper
χ-continuous. Roughly speaking, in L upper χ-continuity is additive. The
following question arises naturally: is χ-completeness an additive property
of complemented modular lattices? It follows from Corollary 1 to Theorem 1
below that the answer to this question is in the negative.A complemented modular lattice is called a Von Neumann geometry or
continuous geometry if it is complete and continuous. In particular a
complete Boolean algebra is a Von Neumann geometry. In any case in a Von
Neumann geometry the set of elements which possess a unique complement form
a complete Boolean algebra. This Boolean algebra is called the centre of the
Von Neumann geometry. Theorem 2 shows that any complete Boolean algebra can
be the centre of a Von Neumann geometry with a homogeneous basis of order n
(see [3] Part II, definition 3.2 for the definition of a homogeneous basis),
n being any fixed natural integer.
Homogeneous Basis for Continuous Geometry
