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Partial Solution to Mackey's Problem about Modular Pairs and Completeness

Published online by Cambridge University Press:  20 November 2018

Samuel S. Holland Jr.
Samuel S. Holland Jr.*
Affiliation:
University of Massachusetts, Amherst, Massachusetts
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Two elements A, B of a lattice are said to form a modular pair when (X ∨ A) Λ B = X ∨ (A Λ B) holds for all X ≦ B, and are said to form a dual-modular pair when (X Λ A) ∨ B = X Λ (A ∨ B) holds for all X ≧ B.

We are concerned here with a particular lattice, the lattice of closed subspaces of a normed linear space, and with a question posed by Mackey in 1945 (6, p. 206, problem 2), namely:

“Are there any incomplete normed linear spaces in whose lattices of closed subspaces modularity and d-modularity are equivalent?”.

The principal result of this paper is the following.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1518 - 1525
Copyright
Copyright © Canadian Mathematical Society 1969

References

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