Chains in generalized Boolean lattices

Abstract

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 = C φ 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.