Abstract
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.
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References
Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer Academic Publishers, Dordrecht
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 representations. Int J Theor Phys 40:703–726
Dvurečenskij A, Vetterlein T (2001c) Generalized pseudo-effect algebras. In: Lectures on soft computing and fuzzy logic. Adv Soft Comput, pp 89–111
Dvurečenskij A, Vetterlein T (2003) Infinitary lattice and Riesz properties of pseudo effect algebras and po-groups. J Aust Math Soc 75:295–312
Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Int J Theor Phys 24:1331–1352
Foulis DJ, Pulmannová S (2015) Unitizing a generalized pseudo effect algebra. Order 32:189–204
Foulis DJ, Pulmannová S, Vinceková E (2014) The exocenter and type decomposition of a generalized pseudo effect algebras. Math Phys 33:13–47
Gudder S, Pulmannová S (1997) Quotients of partial abelian monoids. Algebra Universalis 38:395–421
Jenča G, Pulmannová S (2002) Qutients of partial abelian monoids and the Riesz decomposition property. Algebra Universalis 47:443–477
Pulmannová S (1999) Effect algebras with the Riesz decomposition property and AF C*-algebras. Found Phys 29:1389–1401
Riečanová Z (1999) Subalgebras, intervals, and central elements of generalized effect algebras. Int J Theor Phys 38:3209–3220
Shang Y (2005) The research of effect algebra and pseudo effect algebra in quantum logic. Doctoral Dissertation, Shaanxi Normal University (in chinese)
Shang Y (2008) Direct limit of pseudo effect algebras $\ast $. In: Natural computation, 2008 fourth international conference on Jinan, Shandong, China, pp 309–313
Xie Y, Li Y (2010) Riesz ideals in generalized pseudo effect algebras and in their unitizations. Soft Comput 14:387–398
Xie Y, Li Y, Guo J, Ren F, Li De-chao (2011) Weak commutative pseudo-effect algebras. Int J Theor Phys 50:1186–1197
Acknowledgements
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)
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Communicated by A. Di Nola.
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Guo, Y., Xie, Y. Direct limits of generalized pseudo-effect algebras with the Riesz decomposition properties. Soft Comput 23, 1071–1078 (2019). https://doi.org/10.1007/s00500-018-3121-1
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DOI: https://doi.org/10.1007/s00500-018-3121-1