Skip to main content
SpringerLink
Log in

How to Produce S-Tense Operators on Lattice Effect Algebras

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

Tense operators in effect algebras play a key role for the representation of the dynamics of formally described physical systems. For this, it is important to know how to construct them on a given effect algebra \( E\) and how to compute all possible pairs of tense operators on \( E\). However, we firstly need to derive a time frame which enables these constructions and computations. Hence, we usually apply a suitable set of states of the effect algebra \( E\) in question. To approximate physical reality in quantum mechanics, we use only the so-called Jauch–Piron states on \( E\) in our paper. To realize our constructions, we are restricted on lattice effect algebras only.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Belluce, L.P., Grigolia, R., Lettieri, A.: Representations of monadic MV-algebras. Stud. Log. 81, 123–144 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Botur, M., Paseka, J.: On tense MV-algebras. Fuzzy Set Syst (2014). doi:10.1016/j.fss.2014.06.006

  3. Burges, J.: Basic tense logic. In: Gabbay, D.M., Günther, F. (eds.) Handbook of Philosophical Logic, vol. II, pp. 89–139. D. Reidel Publ. Comp. (1984)

  4. Butnariu, D., Klement, E.P.: Triangular Norm-Based Measures and Games with Fuzzy Coalitions. Kluwer Academic Publishers, Dordrecht (1993)

    Book  MATH  Google Scholar 

  5. Cattaneo, G., Ciucci, D., Dubois, D.: Algebraic models of deviant modal operators based on de Morgan and Kleene lattices. Inf. Sci. 181, 4075–4100 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chajda, I., Kolařík, M.: Dynamic effect algebras. Math. Slovaca 62, 379–388 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chajda, I., Paseka, J.: Dynamic effect algebras and their representations. Soft Comput. 16, 1733–1741 (2012)

    Article  MATH  Google Scholar 

  8. Chajda, I., Paseka, J.: Tense operators and dynamic De Morgan algebras, In: Proceedings of 2013 IEEE 43rd International Symposium Multiple-Valued Logic, pp. 219–224. Springer, Berlin (2013)

  9. Cignoli, R.L.O., D’ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  10. Diaconescu, D., Georgescu, G.: Tense operators on MV-algebras and Łukasiewicz-Moisil algebras. Fundam. Inf. 81, 379–408 (2007)

    MATH  MathSciNet  Google Scholar 

  11. Di Nola, A., Navara, M.: The \(\sigma \)-complete MV-algebras which have enough states. Colloq. Math. 103, 121–130 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  13. Dvurečenskij, A.: Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups. J. Aust. Math. Soc. 82, 183–207 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Figallo, A.V., Pelaitay, G.: Tense operators on De Morgan algebras. Logic J. IGPL (2013). doi:10.1093/jigpal/jzt024

  15. Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1325–1346 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  16. Paseka, J., Janda, J.: A dynamic effect algebras with dual operation. Math. Appl. 1, 79–89 (2012)

    MATH  Google Scholar 

  17. Paseka, J.: Operators on MV-algebras and their representations. Fuzzy Sets Syst. 232, 62–73 (2013). doi:10.1016/j.fss.2013.02.010

    Article  MathSciNet  Google Scholar 

  18. PulmannovÁ, S., Pták, P.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht (1991)

    MATH  Google Scholar 

  19. Riečanová, Z.: Generalization of blocks for D-Lattices and lattice-ordered effect algebras. Int. J. Theor. Phys. 39, 231–237 (2000)

    Article  MATH  Google Scholar 

  20. Riečanová, Z.: Basic decomposition of elements and Jauch–Piron effect algebras. Fuzzy Sets Syst. 155, 138–149 (2005)

    Article  MATH  Google Scholar 

  21. Rutledge, J.D.: A preliminary investigation of the in nitely many-valued predicate calculus. Ph.D. thesis, Cornell University (1959)

  22. Teheux, B.: A Duality for the Algebras of a Łukasiewicz \(n+1\)-valued modal system. Stud. Log. 87, 13–36 (2007). doi: 10.1007/s11225-007-9074-5

    Article  MATH  MathSciNet  Google Scholar 

  23. Teheux B.: Algebraic approach to modal extensions of Łukasiewicz logics. Doctoral thesis, Université de Liege. http://orbi.ulg.ac.be/handle/2268/10887 (2009)

Download references

Acknowledgments

We thank the anonymous referees for the careful reading of the paper and the suggestions on improving its presentation. All authors acknowledge the support by ESF Project CZ.1.07/2.3.00/20.0051 Algebraic methods in Quantum Logic of the Masaryk University.

Author information

Authors and Affiliations

  1. Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46 , Olomouc, Czech Republic

    Ivan Chajda

  2. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 , Brno, Czech Republic

    Jiří Janda & Jan Paseka

Authors
  1. Ivan Chajda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiří Janda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Paseka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Paseka.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chajda, I., Janda, J. & Paseka, J. How to Produce S-Tense Operators on Lattice Effect Algebras. Found Phys 44, 792–811 (2014). https://doi.org/10.1007/s10701-014-9818-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-014-9818-9

Keywords

Mathematics Subject Classification

Search

Navigation