Advertisement

Soft Computing

, Volume 21, Issue 17, pp 4981–4994 | Cite as

Lexicographic pseudo effect algebras

  • Anatolij DvurečenskijEmail author
Foundations
  • 121 Downloads

Abstract

We characterize effect algebras and pseudo effect algebras that can be represented as an interval of the lexicographic product of an antilattice unital po-group (Hu) with a directed po-group G, both with the Riesz Decomposition Property (RDP). We show that a crucial condition is the existence of a lexicographic normal ideal. Finally, we present a categorical equivalence of the category of (Hu)-lexicographic pseudo effect algebras having RDP with the category of directed po-groups with RDP. In addition, a weaker form of the (Hu)-lexicographic pseudo effect algebra is studied.

Keywords

Effect algebra Pseudo effect algebra The Riesz Decomposition Property Po-group Strong unit Lexicographic product Retractive ideal Lexicographic ideal Antilattice \((H, u)\)-perfect effect algebra Lexicographic effect algebra Strongly \((H, u)\)-perfect effect algebra 

Notes

Acknowledgments

This study was funded by the Grant VEGA No. 2/0069/16 SAV and GAČR 15-15286S.

Compliance with ethical standards

Conflicts of interest

Author declares that he has no conflict of interest.

Human participants or animals

This article does not contain any studies with human participants or animals performed by the author.

References

  1. Birkhoff G (1967) Lattice theory. Am Math Soc Coll Publ, vol 25, Providence, Rhode IslandGoogle Scholar
  2. Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann Math 37:823–843MathSciNetCrossRefzbMATHGoogle Scholar
  3. Darnel MR (1995) Theory of lattice-ordered groups. Marcel Dekker Inc, New YorkzbMATHGoogle Scholar
  4. Di Nola A, Grigolia R (2015) Gödel spaces and perfect MV-algebras. J Appl Logic 13:270–284MathSciNetCrossRefzbMATHGoogle Scholar
  5. Di Nola A, Lettieri A (1994) Perfect MV-algebras are categorically equivalent to abelian \(\ell \)-groups. Stud Logica 53:417–432MathSciNetCrossRefzbMATHGoogle Scholar
  6. Di Nola A, Lettieri A (1996) Coproduct MV-algebras, nonstandard reals and Riesz spaces. J Algebra 185:605–620MathSciNetCrossRefzbMATHGoogle Scholar
  7. Diaconescu D, Flaminio T, Leuştean I (2014) Lexicographic MV-algebras and lexicographic states. Fuzzy Sets Syst 244:63–85. doi: 10.1016/j.fss.2014.02.010 MathSciNetCrossRefzbMATHGoogle Scholar
  8. Dvurečenskij A (2002) Pseudo MV-algebras are intervals in \(\ell \)-groups. J Aust Math Soc 72:427–445MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dvurečenskij A (2003) Ideals of pseudo-effect algebras and their applications. Tatra Mt Math Publ 27:45–65MathSciNetzbMATHGoogle Scholar
  10. Dvurečenskij A (2007) Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups. J Aust Math Soc 82:183–207MathSciNetCrossRefzbMATHGoogle Scholar
  11. Dvurečenskij A (2015) Lexicographic pseudo MV-algebras. J Appl Logic 13:825–841. doi: 10.1016/j.jal.2015.10.001 MathSciNetCrossRefzbMATHGoogle Scholar
  12. Dvurečenskij A (2016a) Lexicographic effect algebras. Algebra Univers 75:451–480. doi: 10.1007/s00012-016-0374-3
  13. Dvurečenskij A (2016b) Riesz decomposition properties and the lexicographic product of po-groups. Soft Comput 20:2103–2117. doi: 10.1007/s00500-015-1903-2
  14. Dvurečenskij A, Vetterlein T (2001) Pseudoeffect algebras. I. Basic properties. Int J Theor Phys 40:685–701MathSciNetCrossRefzbMATHGoogle Scholar
  15. Dvurečenskij A, Vetterlein T (2001) Pseudoeffect algebras. II. Group representation. Int J Theor Phys 40:703–726MathSciNetCrossRefzbMATHGoogle Scholar
  16. Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1331–1352MathSciNetCrossRefzbMATHGoogle Scholar
  17. Fuchs L (1963) Partially ordered algebraic systems. Pergamon Press, OxfordzbMATHGoogle Scholar
  18. Georgescu G (2001) Iorgulescu A. Pseudo-MV algebras. Multi Valued Logic 6:95–135MathSciNetzbMATHGoogle Scholar
  19. Glass AMW (1999) Partially ordered groups. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar
  20. Goodearl KR (1986) Partially Ordered Abelian Groups with Interpolation. Math. Surveys Monographs No. 20, Am. Math. Soc., Providence, Rhode IslandGoogle Scholar
  21. Hájek P (2003) Observations on non-commutative fuzzy logic. Soft Comput 8:38–43CrossRefzbMATHGoogle Scholar
  22. Luxemburg WAJ, Zaanen AC (1971) Riesz spaces I. North-Holland, AmsterdamzbMATHGoogle Scholar
  23. Mac Lane S (1971) Categories for the working mathematician. Springer, New YorkCrossRefzbMATHGoogle Scholar
  24. Pulmannová S (2002) Compatibility and decomposition ofeffects. J Math Phys 43:2817–2830MathSciNetCrossRefzbMATHGoogle Scholar
  25. Rachůnek J (2002) A non-commutative generalization of MV-algebras. Czechoslov Math J 52:255–273MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia
  2. 2.Department of Algebra and GeometryPalacký UniversityOlomoucCzech Republic

Personalised recommendations