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A Finiteness Criterion for Orthomodular Lattices

Published online by Cambridge University Press:  20 November 2018

Günter Bruns
Günter Bruns*
Affiliation:
McMaster University, Hamilton, Ontario
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The main result of this paper is the following:

THEOREM. Every finitely generated orthomodular lattice L with finitely manymaximal Boolean subalgebras (blocks) is finite.

If L has one block only, our theorem reduces to the well-known fact that every finitely generated Boolean algebra is finite. On the other hand, it is known that a finitely generated orthomodular lattice without any further restrictions can be infinite. In fact, in [2] we constructed an orthomodular lattice which is generated by a three-element set with two comparable elements, has infinitely many blocks and contains an infinite chain.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 2 , 01 April 1978 , pp. 315 - 320
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Birkhoff, G., Lattice theory, Amer. Math. Soc. Coll. Publ. XXV (Amer. Math. Soc, Providence, 1967).Google Scholar
2. Bruns, G. and Kalmbach, G., Some remarks on free orthomodular lattices, Proc. Univ. Houston Lattice Theory Conf., Houston (1973), 397–408.Google Scholar
3. Foulis, D. J., A note on orthomodular lattices, Portugal. Math. 21 (1962), 65–72.Google Scholar
4. Holland, S. S., A Radon-Nikodym theorem for dimension lattices, Trans. Amer. Math. Soc. 108 (1963), 66–87.Google Scholar