Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-15T07:15:13.513Z Has data issue: false hasContentIssue false
  • English
  • Français

Varieties of Orthomodular Lattices

Published online by Cambridge University Press:  20 November 2018

Günter Bruns  and
Gudrun Kalmbach
Günter Bruns
Affiliation:
McMaster University, Hamilton, Ontario; University of Massachusetts, Amherst, Massachusetts
Gudrun Kalmbach
Affiliation:
Pennsylvania State University, University Park, Pennsylvania University of Massachusetts, Amherst, Massachusetts
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.

In this paper we start investigating the lattice of varieties of orthomodular lattices. The varieties studied here are those generated by orthomodular lattices which are the horizontal sum of Boolean algebras. It turns out that these form a principal ideal in the lattice of all varieties of orthomodular lattices. We give a complete description of this ideal; in particular, we show that each variety in it is generated by its finite members. We furthermore show that each of these varieties is finitely based by exhibiting a (rather complicated) finite equational basis for each variety.

Our methods rely heavily on B. Jonsson's fundamental results in [8]. This, however, could be avoided by starting out with the equations given in sections 3 and 4. Some of our arguments were suggested by Baker [1],

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 5 , October 1971 , pp. 802 - 810
Copyright
Copyright © Canadian Mathematical Society 1971

Footnotes

The research of the firstnamed author was supported by NRC Grant 214-1518.

References

1. Baker, K., Equational classes of modular lattices, Pacific J. Math. 28 (1969), 9–15.Google Scholar
2. Birkhoff, G., Lattice theory, Amer. Math. Soc. Coll. Publ. XXV (Amer. Math. Soc, Providence, 1967).Google Scholar
3. Foulis, D. J., A note on orthomodular lattices, Portugal. Math. 21 (1962), 65–72.Google Scholar
4. Grätzer, G., Universal algebra (van Nostrand, New York, 1968).Google Scholar
5. Greechie, R. J., On the structure of orthomodular lattices satisfying the chain condition, J. Combinatorial Theory 4 (1968), 210–218.Google Scholar
6. Higman, G., Ordering by divisibility in abstract algebras, Proc. London Math. Soc. 2 (1952), 326–336.Google Scholar
7. Holland, S. S., A Radon-Nikodym theorem for dimension lattices, Trans. Amer. Math. Soc. 108 (1963), 66–87.Google Scholar
8. Jönsson, B., Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110–121.Google Scholar
9. Holland, S. S., Equational classes of lattices, Math. Scand. 22 (1968), 187–196.Google Scholar
10. MacLaren, M. D., Nearly modular orthocomplemented lattices, Boeing Sci. Res. Lab. D 1-82-0363 (1964).Google Scholar