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The orthomodular identity and metric completeness of the coordinatizing division ring


Author: Ronald P. Morash
Journal: Proc. Amer. Math. Soc. 27 (1971), 446-448
MSC: Primary 06.40; Secondary 81.00
Erratum: Proc. Amer. Math. Soc. 29 (1971), 627.
Erratum: Proc. Amer. Math. Soc. 29 (1971), 627.
MathSciNet review: 0272689
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ F$ be any division subring of the real quaternions $ H$. Let $ {l_2}(F)$ denote the linear space of all square summable sequences from $ F$ and let $ L$ denote the lattice of all ``$ \bot $-closed'' subspaces of $ {l_2}(F)$, where ``$ \bot $'' denotes the orthogonality relation derived from the $ H$-valued form $ (a,b) = \sum\nolimits_{i = 1}^\infty {{a_i}{{\overline b }_i}} $ where $ a,b \in {l_2}(F),a = ({a_i};i = 1,2, \cdots )$ and $ b = ({b_i};i = 1,2, \cdots )$. Then $ L$ is complete, orthocomplemented, $ M$-symmetric, irreducible, atomistic, and separable, but $ L$ is orthomodular if and only if $ F$ is either the reals, the complex numbers, or the quaternions.


References [Enhancements On Off] (What's this?)

  • [1] Garrett Birkhoff, Lattice theory, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
  • [2] Samuel S. Holland Jr., The current interest in orthomodular lattices, Trends in Lattice Theory (Sympos., U.S. Naval Academy, Annapolis, Md., 1966), Van Nostrand Reinhold, New York, 1970, pp. 41–126. MR 0272688
  • [3] S. Maeda, Theory of symmetric lattices, University of Massachusetts, Amherst, Mass., 1968 (lecture notes--unpublished).
  • [4] M. Donald MacLaren, Atomic orthocomplemented lattices, Pacific J. Math. 14 (1964), 597–612. MR 0163860
  • [5] -, Notes on axioms for quantum mechanics, Argonne National Lab. Report ANL 7065, July 1965.
  • [6] Neal Zierler, Axioms for non-relativistic quantum mechanics, Pacific J. Math. 11 (1961), 1151–1169. MR 0140972
  • [7] George W. Mackey, The mathematical foundations of quantum mechanics: A lecture-note volume, W., A. Benjamin, Inc., New York-Amsterdam, 1963. MR 0155567
  • [8] Samuel S. Holland Jr., Partial solution to Mackey’s problem about modular pairs and completeness, Canad. J. Math. 21 (1969), 1518–1525. MR 0253025

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0272689-3
Keywords: Orthomodular lattices, coordinatization
Article copyright: © Copyright 1971 American Mathematical Society