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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] 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
  • [2] Roman Sikorski, Boolean algebras, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, NeueFolge, Band 25, Academic Press Inc., New York; Springer-Verlag, Berlin-New York, 1964. MR 0177920
  • [3] Neal Zierler, Axioms for non-relativistic quantum mechanics, Pacific J. Math. 11 (1961), 1151–1169. MR 0140972

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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