Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-01T03:59:29.234Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Examples of Complemented Modular Lattices

Published online by Cambridge University Press:  20 November 2018

G. Grätzer  and
Maria J. Wonenburger
G. Grätzer
Affiliation:
Hungarian Academy, Queen's University
Maria J. Wonenburger
Affiliation:
Hungarian Academy, Queen's University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 5 , Issue 2 , May 1962 , pp. 111 - 121
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Amemiya, I. and Halperin, I., Complemented modular lattices. Can. J. Math. Vol. 11, pp. 481-520.Google Scholar
2. Fryer, K. D. and Halperin, I., The von Neumann coordinatization theorem for complemented modular lattices. Acta Szeged 1956, pp. 203-249.Google Scholar
3. J. von Neumann, , Continuous Geometry. Princeton University Press, 1960.Google Scholar