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Chains in generalized Boolean lattices

Published online by Cambridge University Press:  09 April 2009

Richard D. Byrd  and
Roberto A. Mena
Richard D. Byrd
Affiliation:
Department of MathematicsUniversity of Houston, Texas 77004, U.S.A.
Roberto A. Mena
Affiliation:
Department of MathematicsUniversity of Houston, Texas 77004, U.S.A.
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A chain C in a distributive lattice L is called strongly maximal in L if and only if for any homomorphism φ of L onto a distributive lattice K, the chain (Cφ)0 is maximal in K, where (Cφ)0 = Cφ if 0 ∉ K, and (Cφ)0 = Cφ ∪ {0}, otherwise. Gratzer (1971, Theorem 28) states that if B is a generalized Boolean lattice R-generated by L and C is a chain in L, then C R-generates B if and only if C is strongly maximal in L. In this note (Theorem 4.6), we prove the following assertion, which is not far removed from Gratzer's statement: let B be a generalized Boolean lattice R-generated by L and C be a chain in L. If 0 ∈ L, then C generates B if and only if C is strongly maximal in L. If 0 ∉ L, then C generates B if and only if C is strongly maximal in L and [C)L = L. In Section 5 (Example 5.1) a counterexample to Gratzer's statement is provided.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 2 , March 1976 , pp. 234 - 240
Copyright
Copyright © Australian Mathematical Society 1976

References

Byrd, R. D., Mena, R. A. and Troy, L. A. (to appear), ‘Generalized Boolean Lattices’, J. Austral. Math. Soc.Google Scholar
Gratzer, G. (1971), Lattice Theory (W. H. Freeman and Company, San Francisco, 1971).Google Scholar
Makinson, D. C. (1969), ‘On the number of ultrafilters in a Boolean algebra’, Z. Math. Logik Grundlagen Math. 15, 121–122.CrossRefGoogle Scholar