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A remark on λ -regular orthomodular lattices

Vladimír Rogalewicz — 1989

Aplikace matematiky

A finite orthomodular lattice in which every maximal Boolean subalgebra (block) has the same cardinality k is called λ -regular, if each atom is a member of just λ blocks. We estimate the minimal number of blocks of λ -regular orthomodular lattices to be lower than of equal to λ 2 regardless of k .

Orthomodular lattices with almost orthogonal sets of atoms

Sylvia Pulmannová;  Vladimír Rogalewicz — 1991

Commentationes Mathematicae Universitatis Carolinae

The set A of all atoms of an atomic orthomodular lattice is said to be almost orthogonal if the set { b ∈ A : b ≰ a ' } is finite for every a ∈ A . It is said to be strongly almost orthogonal if, for every a ∈ A , any sequence b 1 , b 2 , ⋯ of atoms such that a ≰ b 1 ' , b 1 ≰ b 2 ' , ⋯ contains at most finitely many distinct elements. We study the relation and consequences of these notions. We show among others that a complete atomic orthomodular lattice is a compact topological one if and only if the set of all its atoms is almost orthogonal.

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