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Varieties of Orthomodular Lattices. II

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
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In this paper we continue the study of equationally defined classes of orthomodular lattices started in [1].

The only atom in the lattice of varieties of orthomodular lattices is the variety of all Boolean algebras. Every nontrivial variety contains it. It follows from B. Jónsson [4, Corollary 3.2] that the variety [MO2] generated by the orthomodular lattice MO2 of Figure 1 covers the variety of all Boolean algebras. I t was first shown by R. J. Greechie (oral communication) and is not difficult to see that every variety not consisting of Boolean algebras only contains [MO2]. Again it follows from the result of Jónsson's mentioned above that the varieties generated by one of the orthomodular lattices of Figures 2 to 5 cover [MO2]. The Figures 4 and 5 are to be understood in such a way that the orthocomplement of every element is on the vertical line through this element.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 2 , April 1972 , pp. 328 - 337
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Bruns, G. and Kalmbach, G., Varieties of orthomodular lattices, Can. J. Math. 23 (1971), 802–810.Google Scholar
2. Greechie, R. J., On the structure of orthomodular lattices satisfying the chain condition, J. Combinatorial Theory 4 (1968), 210–218.Google Scholar
3. Greechie, R. J., Orthomodular lattices admitting no states, J. Combinatorial Theory 10 (1971), 119–132.Google Scholar
4. Jónsson, B., Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110–121.Google Scholar