Hyper-operations in effect algebras



The structure of effect algebras is studied from the perspective of hyper-operations. It is shown that Riesz congruences are compatible with the hyper-meet operation and the hyper-join operation in effect algebras with the maximality property. Moreover, we prove that the quotient of an effect algebra E with the maximality property by a Riesz ideal \(I\ne E\) has the maximality property.


Effect algebra Maximality property Hyper-operation Riesz congruence 



The author is highly grateful to the editor and the anonymous referees for their valuable comments and suggestions.


This study was funded by National Natural Science Foundation of China (Grant No. 11401128) and Doctoral Starting up Foundation of Guilin University of Technology.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

Ethical approval

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


  1. Avallone A, Vitolo P (2003) Congruences and ideals of effect algebras. Order 20(1):67–77MathSciNetCrossRefMATHGoogle Scholar
  2. Bennett MK, Foulis DJ (1995) Phi-symmetric effect algebras. Found Phys 25(12):1699–1722MathSciNetCrossRefGoogle Scholar
  3. Bennett MK, Foulis DJ (1998) A generalized Sasaki projection for effect algebras. Tatra Mt Math Publ 15:55–66MathSciNetMATHGoogle Scholar
  4. Dvurečenskij A, Hyčko M (2017) Hyper effect algebras. Fuzzy Sets Syst 326:34–51MathSciNetCrossRefMATHGoogle Scholar
  5. Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer, DordrechtCrossRefMATHGoogle Scholar
  6. Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24(10):1331–1352MathSciNetCrossRefMATHGoogle Scholar
  7. Gudder S, Pulmannová S (1997) Quotients of partial abelian monoids. Algebra Univers 38(4):395–421MathSciNetCrossRefMATHGoogle Scholar
  8. Jenča G (2001) Blocks of homogeneous effect algebras. Bull Aust Math Soc 64(1):81–98MathSciNetCrossRefMATHGoogle Scholar
  9. Jenča G (2003) Finite homogeneous and lattice ordered effect algebras. Discrete Math 272:197–214MathSciNetCrossRefMATHGoogle Scholar
  10. Jenča G (2010) Sharp and meager elements in orthocomplete homogeneous effect algebras. Order 27(1):41–61MathSciNetCrossRefMATHGoogle Scholar
  11. Jenča G, Pulmannová S (2001) Ideals and quotients in lattice ordered effect algebras. Soft Comput 5(5):376–380CrossRefMATHGoogle Scholar
  12. Jenča G, Pulmannová S (2003) Orthocomplete effect algebras. Proc Am Math Soc 131(9):2663–2671MathSciNetCrossRefMATHGoogle Scholar
  13. Jenča G, Riečanová Z (1999) On sharp elements in lattice ordered effect algebras. BUSEFAL 80:24–29Google Scholar
  14. Kôpka F (1992) D-posets of fuzzy sets. Tatra Mt Math Publ 1:83–87MathSciNetMATHGoogle Scholar
  15. Marty F (1934) Sur une generalization de la notion de groupe. In: 8th congres des Mathematiciens Scandinaves, Stockholm, pp 45–49Google Scholar
  16. Pulmannová S, Vinceková E (2007) Riesz ideals in generalized effect algebras and in their unitizations. Algebra Univers 57(4):393–417MathSciNetCrossRefMATHGoogle Scholar
  17. Tkadlec J (2009) Effect algebras with the maximality property. Algebra Univers 61(2):187–194MathSciNetCrossRefMATHGoogle Scholar
  18. Tkadlec J (2017) Properties of effect algebras based on sets of upper bounds. Int J Theor Phys 56(12):4133–4142MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceGuilin University of TechnologyGuilinChina

Personalised recommendations