Categorical Equivalence Between Orthomodular Dynamic Algebras and Complete Orthomodular Lattices

  • Kohei Kishida
  • Soroush Rafiee Rad
  • Joshua Sack
  • Shengyang Zhong


This paper provides a categorical equivalence between two types of quantum structures. One is a complete orthomodular lattice, which is used for reasoning about testable properties of a quantum system. The other is an orthomodular dynamic algebra, which is a quantale used for reasoning about quantum actions. The result extends to more restrictive lattices than orthomodular lattices, and includes Hilbert lattices of closed subspaces of a Hilbert space. These other lattice structures have connections to a wide range of different quantum structures; hence our equivalence establishes a categorical connection between quantales and a great variety of quantum structures.


Orthomodular dynamic algebra Complete orthomodular lattice Quantale 


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK
  2. 2.Institute for Logic, Language and ComputationUniversity of AmsterdamAmsterdamThe Netherlands
  3. 3.Department of Mathematics and StatisticsCalifornia State University Long BeachLong BeachUSA
  4. 4.Institute of Logic and CognitionSun Yat-sen UniversityGuangzhouChina

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