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Keywords:
non-classical logics; D-posets; effect algebras; $MV$-algebras; interval and order topology; states
non-classical logics; D-posets; effect algebras; $MV$-algebras; interval and order topology; states
Summary:
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 a state smearing theorem for these effect algebras, as well as the continuity of $øplus$-operation in the order and interval topologies on them.
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