Kybernetika 46 no. 6, 953-970, 2010

Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras

Jan Paseka, Zdenka Riečanová and Junde Wu


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 $\oplus$-operation in the order and interval topologies on them.


non-classical logics, effect algebras, states, D-posets, $MV$-algebras, interval and order topology


03G12, 06F05, 03G25, 54H12, 08A55