Soft Computing

, Volume 23, Issue 4, pp 1071–1078 | Cite as

Direct limits of generalized pseudo-effect algebras with the Riesz decomposition properties

  • Yanan Guo
  • Yongjian XieEmail author


In this paper, we focus on direct limits and inverse limits in the category with generalized pseudo-effect algebras (GPEAs for short) as objects and GPEA-morphisms as morphisms. We show that direct limits exist in the category of GPEAs and direct limits of GPEAs satisfy the Riesz decomposition properties whenever the directed systems of GPEAs satisfy the Riesz decomposition properties. Then, we give a condition under which the quotient of a direct limit of GPEAs is a direct limit of quotients of GPEAs. Moreover, we prove that if inverse systems of GPEAs satisfy the Riesz decomposition properties, then inverse limits also satisfy the Riesz decomposition properties.


Generalized pseudo-effect algebras Riesz decomposition properties Direct limits Inverse limits 



This article does not contain any studies with human participants or animals performed by any of the authors. Informed consent was obtained from all individual participants included in the study. The authors are grateful to the anonymous referee’s valuable and constructive comments. This work is partially by National Science Foundation of China (Grant Nos. 61673250, 11201279) and the Fundamental Research Funds for the Central Universities (Grant No. GK201503017)

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  2. Dvurečenskij A, Vetterlein T (2001a) Pseudoeffect algebras. I. Basic properties. Int J Theor Phys 40:685–701MathSciNetCrossRefzbMATHGoogle Scholar
  3. Dvurečenskij A, Vetterlein T (2001b) Pseudoeffect algebras. II. Group representations. Int J Theor Phys 40:703–726MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dvurečenskij A, Vetterlein T (2001c) Generalized pseudo-effect algebras. In: Lectures on soft computing and fuzzy logic. Adv Soft Comput, pp 89–111Google Scholar
  5. Dvurečenskij A, Vetterlein T (2003) Infinitary lattice and Riesz properties of pseudo effect algebras and po-groups. J Aust Math Soc 75:295–312MathSciNetCrossRefzbMATHGoogle Scholar
  6. Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Int J Theor Phys 24:1331–1352MathSciNetzbMATHGoogle Scholar
  7. Foulis DJ, Pulmannová S (2015) Unitizing a generalized pseudo effect algebra. Order 32:189–204MathSciNetCrossRefzbMATHGoogle Scholar
  8. Foulis DJ, Pulmannová S, Vinceková E (2014) The exocenter and type decomposition of a generalized pseudo effect algebras. Math Phys 33:13–47zbMATHGoogle Scholar
  9. Gudder S, Pulmannová S (1997) Quotients of partial abelian monoids. Algebra Universalis 38:395–421MathSciNetCrossRefzbMATHGoogle Scholar
  10. Jenča G, Pulmannová S (2002) Qutients of partial abelian monoids and the Riesz decomposition property. Algebra Universalis 47:443–477MathSciNetCrossRefzbMATHGoogle Scholar
  11. Pulmannová S (1999) Effect algebras with the Riesz decomposition property and AF C*-algebras. Found Phys 29:1389–1401MathSciNetCrossRefGoogle Scholar
  12. Riečanová Z (1999) Subalgebras, intervals, and central elements of generalized effect algebras. Int J Theor Phys 38:3209–3220MathSciNetCrossRefzbMATHGoogle Scholar
  13. Shang Y (2005) The research of effect algebra and pseudo effect algebra in quantum logic. Doctoral Dissertation, Shaanxi Normal University (in chinese) Google Scholar
  14. Shang Y (2008) Direct limit of pseudo effect algebras $\ast $. In: Natural computation, 2008 fourth international conference on Jinan, Shandong, China, pp 309–313Google Scholar
  15. Xie Y, Li Y (2010) Riesz ideals in generalized pseudo effect algebras and in their unitizations. Soft Comput 14:387–398CrossRefzbMATHGoogle Scholar
  16. Xie Y, Li Y, Guo J, Ren F, Li De-chao (2011) Weak commutative pseudo-effect algebras. Int J Theor Phys 50:1186–1197CrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China

Personalised recommendations