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On a new construction of pseudo effect algebras

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Abstract

We define a new class of pseudo effect algebras, called kite pseudo effect algebras, which are connected not necessarily with partially ordered groups, but rather with generalized pseudo effect algebras where the greatest element is not guaranteed. Starting even with a commutative generalized pseudo effect algebra, we can obtain a non-commutative pseudo effect algebra. We show how such kite pseudo effect algebras are tied with different types of the Riesz decomposition properties. We find conditions when kite pseudo effect algebras have the least non-trivial normal ideal.

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References

  • Darnel MR (1995) Theory of lattice-ordered groups. Marcel Dekker Inc., New York

    MATH  Google Scholar 

  • Dvurečcenskij A (2002) Pseudo MV-algebras are intervals in \(\ell \)-groups. J Aust Math Soc 72:427–445

    Article  MathSciNet  Google Scholar 

  • Dvurečenskij A (2002) States on unital partially-ordered groups. Kybernetika 38:297–318

    MATH  MathSciNet  Google Scholar 

  • Dvurečenskij A (2003) Ideals of pseudo-effect algebras and their applications. Tatra Mt Math Publ 27:45–65

    MATH  MathSciNet  Google Scholar 

  • Dvurečenskij A (2013) Kite pseudo effect algebras. Found Phys 43:1314–1338. doi:10.1007/s10701-013-9748-y

    Article  MATH  MathSciNet  Google Scholar 

  • Dvurečenskij A, Holland WCH (2014) Some remarks on kite pseudo effect algebras. Int J Theor Phys 53:1685–1696. doi:10.1007/s10773-013-1966-8

  • Dvurečenskij A, Kowalski T (2014) Kites and pseudo BL-algebras. Algebra Univ 71:235–260. doi:10.1007/s00012-014-0276-1

    Article  MATH  Google Scholar 

  • Dvurečenskij A, Krňávek J (2013) The lexicographic product of po-groups and \(n\)-perfect pseudo effect algebras. Int J Theor Phys 52:2760–2772. doi:10.1007/s10773-013-1568-5

  • Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer Acad. Publ, Ister Science, Dordrecht, Bratislava

  • Dvurečenskij A, Vetterlein T (2001a) Pseudoeffect algebras. I. Basic properties. Int J Theor Phys 40:685–701

  • Dvurečenskij A, Vetterlein T (2001b) Pseudoeffect algebras. II. Group representation. Int J Theor Phys 40:703–726

    Article  MATH  Google Scholar 

  • Dvurečenskij A, Vetterlein T (2001c) Generalized pseudo-effect algebras. In: Di Nola A, Gerla G (eds) Lectures on soft computing and fuzzy logic. Springer, Berlin, pp 89–111

    Chapter  Google Scholar 

  • Dvurečenskij A, Vetterlein T (2001) Congruences and states on pseudo-effect algebras. Found Phys Lett 14:425–446

    Article  MathSciNet  Google Scholar 

  • Dvurečenskij A, Vetterlein T (2002) Algebras in the positive cone of po-groups. Order 19:127–146

    Article  MATH  MathSciNet  Google Scholar 

  • Dvurečenskij A, Vetterlein T (2004) Non-commutative algebras and quantum structures. Int J Theor Phys 43:1599–1612

    Article  MATH  Google Scholar 

  • Dvurečenskij A, Xie Y, Yang A (2013) Discrete \((n+1)\)-valued states and \(n\)-perfect pseudo-effect algebras. Soft Comput 17:1537–1552. doi: 10.1007/s00500-013-1001-2

    Article  Google Scholar 

  • Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1331–1352

    Article  MATH  MathSciNet  Google Scholar 

  • Fuchs L (1963) Partially ordered algebraic systems. Pergamon Press, Oxford, New York

    MATH  Google Scholar 

  • Ravindran K (1996) On a structure theory of effect algebras. Ph.D. thesis, Kansas State University, Manhattan, Kansas

  • Xie Y, Li Y, Guo J, Ren F, Li D (2011) Weak commutative pseudo-effect algebras. Int J Theor Phys 50:1186–1197

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author is very indebted to anonymous referees for their careful reading and suggestions which helped to improve the readability of the paper.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Anatolij Dvurečenskij

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

    Anatolij Dvurečenskij

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  1. Anatolij Dvurečenskij
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Corresponding author

Correspondence to Anatolij Dvurečenskij.

Additional information

Communicated by M. Navara.

This work was supported by the Slovak Research and Development Agency under Contract APVV-0178-11, Grant VEGA No. 2/0059/12 SAV, and CZ.1.07/2.3.00/20.0051.

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Dvurečenskij, A. On a new construction of pseudo effect algebras. Soft Comput 19, 517–529 (2015). https://doi.org/10.1007/s00500-014-1468-5

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  • DOI: https://doi.org/10.1007/s00500-014-1468-5

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