Skip to main content
SpringerLink
Log in

States On Orthocomplemented Difference Posets (Extensions)

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We continue the investigation of orthocomplemented posets that are endowed with a symmetric difference (ODPs). The ODPs are orthomodular and, therefore, can be viewed as “enriched” quantum logics. In this note, we introduced states on ODPs. We derive their basic properties and study the possibility of extending them over larger ODPs. We show that there are extensions of states from Boolean algebras over unital ODPs. Since unital ODPs do not, in general, have to be set-representable, this result can be applied to a rather large class of ODPs. We then ask the same question after replacing Boolean algebras with “nearly Boolean” ODPs (the pseudocomplemented ODPs). Making use of a few results on ODPs, some known and some new, we construct a pseudocomplemented ODP, P, and a state on P that does not allow for extensions over larger ODPs.

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

Similar content being viewed by others

References

  1. Dalla Chiara M.L., Giuntini R., Greechie R.: Reasoning in quantum theory: sharp and unsharp quantum logics. Kluwer, Dordrecht (2004)

    Book  MATH  Google Scholar 

  2. De Simone A., Navara M., Pták P.: States on systems of sets that are closed under symmetric difference. Mathematische Nachrichten. 288(17–18), 1995–2000 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dorfer G., Dvurečenskij A., Länger H.M.: Symmetric difference in orthomodular lattices. Math. Slovaca 46, 435–444 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Dvurečenskij A., Pulmannová S.: New trends in quantum structures. Kluwer Academic Publ., Dordrecht (2000)

    Book  MATH  Google Scholar 

  5. Gudder S.P.: Stochastic methods in quantum mechanics. Dover, Amsterdam (1979)

    MATH  Google Scholar 

  6. Matoušek M.: Orthocomplemented lattices with a symmetric difference. Algebra Univers. 60, 185–215 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Matoušek M., Pták P.: Orthocomplemented posets with a symmetric difference. Order. 26, 1–21 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Matoušek M., Pták P.: On identities in orthocomplemented difference lattices. Math. Slovaca. 60(5), 1–8 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Matoušek M., Pták P.: Orthomodular posets related to Z2-valued states. Int. J. Theor. Phys. 53(10), 3323–3332 (2014)

    Article  MATH  Google Scholar 

  10. Navara M., Pták P.: Almost Boolean orthomodular posets. J. Pure Appl. Algebra. 60, 105–111 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Park E., Kim M.M., Chung J.Y.: A note on symmetric differences of orthomodular lattices. Commun. Korean Math. Soc. 18(2), 207–214 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pták P.: Extensions of states on logics. Bull. Acad. Polon. Sci. Sér. Math. 33, 493–497 (1985)

    MathSciNet  MATH  Google Scholar 

  13. Pták P., Pulmannová S.: Orthomodular structures as quantum logics. Kluwer, Dordrecht (1991)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27, Prague, Czech Republic

    Michal Hroch & Pavel Pták

Authors
  1. Michal Hroch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pavel Pták
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michal Hroch.

Additional information

M. Hroch was supported by the Grant Agency of the Czech Technical University in Prague, Grant No. SGS15/193/OHK3/3T/13.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hroch, M., Pták, P. States On Orthocomplemented Difference Posets (Extensions). Lett Math Phys 106, 1131–1137 (2016). https://doi.org/10.1007/s11005-016-0862-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-016-0862-6

Mathematics Subject Classification

Keywords

Search

Navigation