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Chains in generalized Boolean lattices
Published online by Cambridge University Press: 09 April 2009
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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.
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- 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
Makinson, D. C. (1969), ‘On the number of ultrafilters in a Boolean algebra’, Z. Math. Logik Grundlagen Math. 15, 121–122.CrossRefGoogle Scholar
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