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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Orthomodularity and the direct sum of division subrings of the quaternions

Author: Ronald P. Morash
Journal: Proc. Amer. Math. Soc. 36 (1972), 63-68
MSC: Primary 46C05; Secondary 16A40
MathSciNet review: 0312225
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Abstract: Let D be any division subring of the real quaternions H. Let $ \oplus $ D denote the linear space of all finitely nonzero sequences from D and let L denote the lattice of all ``$ \bot $-closed'' subspaces of $ \oplus $ D, where ``$ \bot $'' denotes the orthogonality relation derived from the H-valued form $ (a,b) = \sum {({a_i}b_i^ \ast :i = 1,2, \cdots )} $ where $ a,b \in \oplus D,a = ({a_1},{a_2}, \cdots ,{a_N},0,0, \cdots )$ and $ b = ({b_1},{b_2}, \cdots ,{b_M},0,0, \cdots )$, and $ b_i^ \ast $ is the quaternionic conjugate of $ {b_i}$. Then, the lattice L is complete and orthocomplemented, but is not orthomodular.

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Keywords: Orthomodular lattices
Article copyright: © Copyright 1972 American Mathematical Society