Soft Computing

, Volume 23, Issue 3, pp 735–745 | Cite as

Congruences and ideals in generalized pseudoeffect algebras revisited

  • S. PulmannováEmail author


The first part of the present paper is an enhancement of the paper (Foulis et al. in Order 33:311–332, 2016). A new type of congruences on generalized pseudoeffect algebras (GPEAs), called R1-congruences, is introduced, which is in one-to-one correspondence with normal R1-ideals. The notion of Riesz congruences is reconsidered, and they are defined as congruences which are in one-to-one correspondence with normal Riesz ideals. In upward directed GPEAs, in particular in pseudoeffect algebras, both these types of congruences as well as ideals coincide. Conditions under which congruences and ideals in a GPEA P may be extended to a \(\gamma \)-unitization of U of P are clarified. In the last part of the paper, subcentral and central ideals in GPEAs and their relations to subdirect and direct decompositions are studied.


Effect algebra Pseudoeffect algebra Generalized pseudoeffect algebra R1-congruence R1-ideal Riesz congruence Riesz ideal Unitization Subcentral ideal 



The author was supported by Research and Development Support Agency under the Contract APVV-16-0073 and Grant VEGA 2/0069/16.

Compliance with ethical standards

Conflict of interest

The author declares that she has no conflict of interests.

Ethical approval

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


  1. Avallone A, Vitolo P (2003) Congruences and ideals of effect algebras. Order 20:67–77MathSciNetCrossRefzbMATHGoogle Scholar
  2. Dvurečenskij A (2003) Ideals of pseudo effect algebras and their applications. Tatra Mt Math Publ 27:45–65MathSciNetzbMATHGoogle Scholar
  3. Dvurečenskij A (2013) Kite pseudo effect algebras. Found Phys 43:1314–1338MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  5. Dvurečenskij A, Vetterlein T (2001a) Pseudo effect algebras I. Basic properties. Int J Theor Phys 40:685–701CrossRefzbMATHGoogle Scholar
  6. Dvurečenskij A, Vetterlein T (2001b) Pseudo effect algebras II. Group representation. Int J Theor Phys 40:703–726CrossRefzbMATHGoogle Scholar
  7. Dvurečenskij A, Vetterlein T (2001c) Generalized pseudo-effect algebras. In: Di Nola A, Gerla G (Eds) Lectures on soft computing and fuzzy logic, Physica-Verlag, Springer, Berlin, pp 89–111Google Scholar
  8. Dvurečenskij A, Vetterlein T (2001d) Congruences and states on pseudo effect algebras. Found Phys Lett 14:425–446MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dvurečenskij A, Vetterlein T (2001e) Generalized pseudoeffect algebras. In: Lectures on soft computing and fuzzy logic. Advances in soft computing, vol 11. Physica, Heidelberg, pp 89–111Google Scholar
  10. Dvurečenskij A, Vetterlein T (2002) Algebras in the positive cone of po-groups. Order 19:127–146MathSciNetCrossRefzbMATHGoogle Scholar
  11. Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1325–1346MathSciNetCrossRefzbMATHGoogle Scholar
  12. Foulis DJ, Pulmannová S (2015) Unitizing a generalized pseudo effect algebra. Order 32:311–332MathSciNetCrossRefzbMATHGoogle Scholar
  13. Foulis DJ, Pulmnnová S, Vinceková E (2013) The exocenter and type decompositions of a generalized pseudoeffect algebra. Discuss Math Gen Algebra Appl 33:13–47MathSciNetCrossRefzbMATHGoogle Scholar
  14. Foulis DJ, Pulmannová S, Vinceková E (2016) Unitizations of generalized pseudoeffect algebras and their ideals. Order 33:311–332MathSciNetCrossRefzbMATHGoogle Scholar
  15. Gudder S, Pulmannová S (1997) Quotients of partial abelian monoids. Algebra Univ 38:395–421MathSciNetCrossRefzbMATHGoogle Scholar
  16. Haiyang Li, Shenggang Li (2008) Congruences and ideals in pseudo effect algebras. Soft Comput 12:487–492CrossRefGoogle Scholar
  17. Jenča G (2000) Subcentral ideals in generalized effect algebras. Int J Theor Phys 19:745–755MathSciNetzbMATHGoogle Scholar
  18. Pulmannová S, Vinceková E (2007) Riesz ideals in generalized effect algebras and in their unitizations. Algebra Univ 57:393–417MathSciNetCrossRefzbMATHGoogle Scholar
  19. Riečanová Z, Marinová I (2005) Generalized homogeneous, prelattice and MV-effect algebras. Kybernetika 41:129–142MathSciNetzbMATHGoogle Scholar
  20. Xie Y, Li Y (2010) Riesz ideals in generalized pseudo effect algebras and in their unitizations. Soft Comput 14:387–398CrossRefzbMATHGoogle Scholar
  21. Xie Y, Li Y, Guo J, Ren F, Li D (2011) Weak commutative pseudo effect algebras. Int J Theor Phys 50:1186–1197CrossRefzbMATHGoogle Scholar

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

Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia

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