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On realization of effect algebras

  • Josef Niederle EMAIL logo and Jan Paseka
Published/Copyright: July 5, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 2

Abstract

A well known fact is that there is a finite orthomodular lattice with an order determining set of states which is not order embeddable into the standard quantum logic, the lattice L(𝓗) of all closed subspaces of a separable complex Hilbert space.

We show that a finite generalized effect algebra is order embeddable into the standard effect algebra E(𝓗) of effects of a separable complex Hilbert space iff it has an order determining set of generalized states iff it is order embeddable into the power of a finite MV-chain.

As an application we obtain an algorithm, which is based on the simplex algorithm, deciding whether such an order embedding exists and, if the answer is positive, constructing it.

MSC 2010: Primary 06C15; Secondary 03G12; 06D35; 06F25; 81P10
Keywords: non-classical logics; orthomodular lattices; effect algebras; MV-algebras; states; simplex algorithm

Dedicated to Anatolij Dvurečenskij on the occassion of his 65th birthday in appreciation of his contributions to the fields of the theory of quantum structures (Communicated by Sylvia Pulmannová)

The authors gratefully acknowledge Financial Support by ESF Project CZ.1.07/2.3.00/20.0051 Algebraic methods in Quantum Logic of the Masaryk University.

J. Paseka acknowledges Financial Support of the Czech Science Foundation (GACR) under the grant Algebraic, many-valued and quantum structures for uncertainty modelling No. GACR 15-15286S.

Acknowledgement

We also thank the anonymous referees for the very thorough reading and contributions to improve our presentation of the paper.

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Received: 2013-12-30
Accepted: 2014-9-3
Published Online: 2016-7-5
Published in Print: 2016-4-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0140
Keywords for this article
non-classical logics; orthomodular lattices; effect algebras; MV-algebras; states; simplex algorithm

Articles in the same Issue

  1. Research Article
  2. Holistic logical arguments in quantum computation
  3. Research Article
  4. States with values in the Łukasiewicz groupoid
  5. Research Article
  6. On realization of effect algebras
  7. Research Article
  8. States on symmetric logics: extensions
  9. Research Article
  10. Extensions and measurability in quantum measure spaces
  11. Research Article
  12. Convex subalgebras of upper bounded GMV-algebras
  13. Research Article
  14. Combining boolean algebras and ℓ-groups in the variety generated by chang’s mv-algebra
  15. Research Article
  16. Large rigid sets of algebras with respect to embeddability
  17. Research Article
  18. Conformal algebras, vertex algebras, and the logic of locality
  19. Research Article
  20. Extension of measures on pseudo-D-lattices
  21. Research Article
  22. Note on a parameter switching method for nonlinear ODEs
  23. Research Article
  24. Compatibility for probabilistic theories
  25. Research Article
  26. Completeness of Gelfand-Neumark-Segal inner product space on Jordan algebras
  27. Research Article
  28. Commutativity in a synaptic algebra
  29. Research Article
  30. Discrete averaged mixing applied to the logarithmic distributions
  31. Research Article
  32. Automorphisms of decompositions
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