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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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: 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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0272689-3
PII: S 0002-9939(1971)0272689-3
Keywords: Orthomodular lattices, coordinatization
Article copyright: © Copyright 1971 American Mathematical Society