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Proceedings of the American Mathematical Society

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

 
 

 

An $ m$-orthocomplete orthomodular lattice is $ m$-complete


Author: Samuel S. Holland
Journal: Proc. Amer. Math. Soc. 24 (1970), 716-718
MSC: Primary 06.40
DOI: https://doi.org/10.1090/S0002-9939-1970-0256949-7
MathSciNet review: 0256949
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Abstract: We call an orthomodular lattice $ \mathcal{L}\;m$-orthocomplete for an infinite cardinal $ m$ if every orthogonal family of $ \leqq m$ elements from $ \mathcal{L}$ has a join in $ \mathcal{L}$, and we call $ \mathcal{L}\;m$-complete if every family, orthogonal or not, of $ \leqq m$ elements from $ \mathcal{L}$ has a join in $ \mathcal{L}$. We prove that an $ m$-orthocomplete orthomodular lattice is $ m$-complete. Since a Boolean algebra is a distributive orthomodular lattice, we obtain as a special case the Smith-Tarski theorem: An $ m$-orthocomplete Boolean algebra is $ m$-complete.


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

  • [1] S. S. Holland, Jr., ``The current interest in orthomodular lattices,'' in Trends in lattice theory, Van Nostrand, Princeton, N. J., 1969. MR 0272688 (42:7569)
  • [2] R. Sikorski, Boolean algebras, 2nd ed., Academic Press, New York and Springer-Verlag, Berlin, 1964. MR 31 #2178. MR 0177920 (31:2178)
  • [3] N. Zierler, Axioms for non-relativistic quantum mechanics, Pacific J. Math. 11 (1961), 1151-1169. MR 25 #4385. MR 0140972 (25:4385)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0256949-7
Keywords: Orthomodular lattices, Boolean algebras, $ m$-orthocomplete, $ m$-complete, Smith-Tarski theorem
Article copyright: © Copyright 1970 American Mathematical Society

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