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Lattices and Quantum Logics with Separated Intervals, Atomicity

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Abstract

It is well known that a Boolean algebra B isatomic (atomistic) iff the interval topology on B isHausdorff. But this no longer holds for orthomodularlattices (quantum logics). There exist (even complete) atomic orthomodular lattices the intervaltopology of which is not Hausdorff. We show that anothercharacterization of atomicity for Boolean algebras isthe following: A Boolean algebra B is atomic iff B has separated intervals. Furthermore, we showthat the interval topology on a complete orthomodularlattice L is Hausdorff iff L has separated intervals iffL is atomic and it has separated intervals. An orthomodular lattice L with orthomodularMacNeille completion \({\hat L}\) has separatedintervals iff L is atomic and it has separated intervalsiff the interval topology on \({\hat L}\) isHausdorff.

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Authors
  1. Zdenka Riecanova
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Riecanova, Z. Lattices and Quantum Logics with Separated Intervals, Atomicity. International Journal of Theoretical Physics 37, 191–197 (1998). https://doi.org/10.1023/A:1026642028987

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  • DOI: https://doi.org/10.1023/A:1026642028987

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