Block-finite atomic orthomodular lattices
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.
compact topological orthomodular lattices
scientific article; zbMATH DE number 445052
scientific article