Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras

Jan Paseka; Zdena Riečanová; Junde Wu

Kybernetika (2010)

  • Volume: 46, Issue: 6, page 953-970
  • ISSN: 0023-5954

Abstract

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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 ø p l u s -operation in the order and interval topologies on them.

How to cite

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Paseka, Jan, Riečanová, Zdena, and Wu, Junde. "Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras." Kybernetika 46.6 (2010): 953-970. <http://eudml.org/doc/196458>.

@article{Paseka2010,
abstract = {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.},
author = {Paseka, Jan, Riečanová, Zdena, Wu, Junde},
journal = {Kybernetika},
keywords = {non-classical logics; D-posets; effect algebras; $MV$-algebras; interval and order topology; states; convergence of nets; effect algebra; interval topology; order topology},
language = {eng},
number = {6},
pages = {953-970},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras},
url = {http://eudml.org/doc/196458},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Paseka, Jan
AU - Riečanová, Zdena
AU - Wu, Junde
TI - Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 6
SP - 953
EP - 970
AB - 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.
LA - eng
KW - non-classical logics; D-posets; effect algebras; $MV$-algebras; interval and order topology; states; convergence of nets; effect algebra; interval topology; order topology
UR - http://eudml.org/doc/196458
ER -

References

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