Applications of coordinatization in modular lattice theory: The legacy of J. von Neumann

Order ◽  
1985 ◽  
Vol 1 (3) ◽  
pp. 295-300 ◽  
Author(s):  
Alan Day
Keyword(s):  
Modular Lattice ◽  
Lattice Theory ◽  
Von Neumann

scholarly journals The completion by cuts of an orthocomplemented modular lattice

1969 ◽  
Vol 1 (2) ◽  
pp. 279-280 ◽  
Author(s):  
D. H. Adams
Keyword(s):  
Modular Lattice ◽  
Lattice Theory

In this note we give an example of an orthocomplemented modular lattice whose completion by cuts is not orthomodular. This solves negatively Problem 36, p. 131, in G. Birkhoff: Lattice theory (3rd edition).

Some Examples of Complemented Modular Lattices

1962 ◽  
Vol 5 (2) ◽  
pp. 111-121 ◽  
Author(s):  
G. Grätzer ◽  
Maria J. Wonenburger
Keyword(s):  
Boolean Algebra ◽  
Modular Lattice ◽  
Complete Boolean Algebra ◽  
Modular Lattices ◽  
Additive Property ◽  
Von Neumann ◽  
Continuous Geometry ◽  
Definition Of ◽  
Homogeneous Basis

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.

Complemented Modular Lattices

1959 ◽  
Vol 11 ◽  
pp. 481-520 ◽  
Author(s):  
Ichiro Amemiya ◽  
Israel Halperin
Keyword(s):  
Modular Lattice ◽  
Modular Lattices ◽  
Von Neumann ◽  
Upper Continuous ◽  
Continuous Geometry

1.1 This paper gives a lattice theoretic investigation of “finiteness“ and “continuity of the lattice operations” in a complemented modular lattice. Although we usually assume that the lattice is-complete for some infinite,3we do not require completeness and continuity, as von Neumann does in his classical memoir on continuous geometry (3); nor do we assume orthocomplementation as Kaplansky does in his remarkable paper (1).1.2. Our exposition is elementary in the sense that it can be read without reference to the literature. Our brief preliminary § 2 should enable the reader to read this paper independently.1.3. Von Neumann's theory of independence (3, Part I, Chapter II) leans heavily on the assumption that the lattice is continuous, or at least upper continuous.

Lectures on von Neumann Algebras

2019 ◽  
Author(s):  
Serban-Valentin Stratila ◽  
Laszlo Zsido
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann

John von Neumann

2004 ◽  
Vol 174 (12) ◽  
pp. 1371 ◽  
Author(s):  
Mikhail I. Monastyrskii
Keyword(s):  
John Von Neumann ◽  
Von Neumann
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