Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T00:58:57.252Z Has data issue: false hasContentIssue false
  • English
  • Français

Block-Finite Orthomodular Lattices

Published online by Cambridge University Press:  20 November 2018

Günter Bruns
Günter Bruns*
Affiliation:
McMaster University, Hamilton, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Introduction. Every orthomodular lattice (abbreviated : OML) is the union of its maximal Boolean subalgebras (blocks). The question thus arises how conversely Boolean algebras can be amalgamated in order to obtain an OML of which the given Boolean algebras are the blocks. This question we deal with in the present paper.

The problem was first investigated by Greechie [6, 7, 8, 9]. His technique of pasting [6] will also play an important role in this paper. A case solved completely by Greechie [9] is the case that any two blocks intersect either in the bounds only or have the bounds, an atom and its complement in common. This is, of course, a very special situation. The more surprising it is that Greechie's methods, if skillfully applied, yield considerable insight into the structure of OMLs and provide a seemingly unexhaustible source for counter-examples.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 5 , 01 October 1979 , pp. 961 - 985
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Birkhoff, G., Lattice theory, Amer. Math. Soc. Coll. Publ. 25 (Amer. Math. Soc, Providence, 1967).Google Scholar
2. Bruns, G., Free ortholattices, Can. J. Math. 28 (1976), 977–985.Google Scholar
3. Bruns, G., Afiniteness criterion for ortho-modular lattices, Can. J. Math. 30 (1978).Google Scholar
4. Bruns, G. and Kalmbach, G., Varieties of orthomodular lattices II, Can. J. Math. 24 (1972), 328–337.Google Scholar
5. Foulis, D. J., A note on orthomodular lattices, Portugal. Math. 21 (1962), 65–72.Google Scholar
6. Greechie, R. J., On the structure of orthomodular lattices satisfying the chain condition, J. Combinatorial Theor. 4 (1968), 210–218.Google Scholar
7. Greechie, R. J., An orthomodular poset with a full set of states not embeddable in Hilbert space, Caribbean J. of Science and Math. 1 (1969).Google Scholar
8. Greechie, R. J., On generating pathological orthomodular structures, Kansas State Univ. Technical report 13 (1970).Google Scholar
9. Greechie, R. J., Orthomodular lattices admitting no states, J. Combinatorial Theor. 10 (1971), 119–132.Google Scholar
10. Holland, S. S., A Radon-Nikodym theorem for dimension lattices, Trans. Amer. Math. Soc. 108 (1963), 66–87.Google Scholar
11. Kalmbach, G., Orthomodular lattices do not satisfy any special lattice equation, Archiv d. Math. 28 (1977), 7–8.Google Scholar
12. MacLaren, M. D., Nearly modular orthocomplemented lattices, Boeing Sci. Res. Lab. D 1-82-0363 (1964).Google Scholar