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Zbl 0786.06007
Pulmannová, S.; Riečanová, Z.
Block-finite atomic orthomodular lattices. (English)
[J] J. Pure Appl. Algebra 89, No.3, 295-304 (1993). ISSN 0022-4049

A block-finite orthomodular lattice $L$ has only finitely many blocks, i.e. maximal Boolean subalgebras. It is shown that the MacNeille completion of $L$ is an orthomodular lattice. Assume $L$ is complete. Then $L$ is atomic iff the interval topology on $L$ is Hausdorff. Let $M$ be a complete, (0)-continuous, commutator-finite orthomodular lattice with trivial center. Then $M$ is atomic. For a complete, (0)-continuous, atomless orthomodular lattice $M$, the conditions: $M$ is a Boolean algebra, $M$ is block-finite, and $M$ is commutator-finite, are equivalent. For compact topological orthomodular lattices $L$ some characterizations are given for $L$ to be profinite, i.e. isomorphic with a direct product of finite orthomodular lattices.
[G.Kalmbach (Ulm)]

MSC 2000:: *06C15 Complemented lattices
06B30 Topological lattices

Keywords: block-finite orthomodular lattice; MacNeille completion; commutator- finite orthomodular lattice; Boolean algebra; compact topological orthomodular lattices; profinite

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