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
- PDF | Request permission
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
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