Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras

Details Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras

2009-08-24

arxiv.org/abs/0908.3288v2

Physics

Mathematical Physics

Scientific paper

We prove that the interval topology of an Archimedean atomic lattice effect algebra $E$ is Hausdorff whenever the set of all atoms of $E$ is almost orthogonal. In such a case $E$ is order continuous. If moreover $E$ is complete then order convergence of nets of elements of $E$ is topological and hence it coincides with convergence in the order topology and this topology is compact Hausdorff compatible with a uniformity induced by a separating function family on $E$ corresponding to compact and cocompact elements. For block-finite Archimedean atomic lattice effect algebras the equivalence of almost orthogonality and s-compact generation is shown. As the main application we obtain the state smearing theorem for these effect algebras, as well as the continuity of $\oplus$-operation in the order and interval topologies on them.

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