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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
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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

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