Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

An $m$-orthocomplete orthomodular lattice is $m$-complete
HTML articles powered by AMS MathViewer

by Samuel S. Holland
Proc. Amer. Math. Soc. 24 (1970), 716-718
DOI: https://doi.org/10.1090/S0002-9939-1970-0256949-7

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
  • 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
  • Roman Sikorski, Boolean algebras, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 25, Academic Press, Inc., New York; Springer-Verlag, Berlin-New York, 1964. MR 0177920
  • Neal Zierler, Axioms for non-relativistic quantum mechanics, Pacific J. Math. 11 (1961), 1151–1169. MR 140972, DOI 10.2140/pjm.1961.11.1151
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 06.40
  • Retrieve articles in all journals with MSC: 06.40
Bibliographic Information
  • © Copyright 1970 American Mathematical Society
  • 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