Abstract
In the paper, we show that the quotient \([E]_I\) of a lattice-ordered pseudoeffect algebra \(E\) with respect to a normal weak Riesz ideal \(I\) is linearly ordered if and only if \(I\) is a prime normal weak Riesz ideal, and \([E]_I\) is a representable pseudo MV-algebra if and only if \(I\) is an intersection of prime normal weak Riesz ideals. Moreover, we introduce the concept of weakly algebraic sets in pseudoeffect algebras, discuss the characterizations of weakly algebraic sets and show that weakly algebraic sets in pseudoeffect algebra \(E\) are in a one-to-one correspondence with normal weak Riesz ideals in pseudoeffect algebra \(E.\)
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References
Avallone A, Vitolo P (2003) Congruences and ideals of effect algebras. Order 20:67–77
Chevalier G, Pulmannová S (2000) Some ideal lattices in partial Abelian monoids and effect algebras. Order 17(1):72–92
Dvurečenskij A (2001) On pseudo-MV algebras. Soft Comput 5:347–354
Dvurečenskij A, Pulmannová, S (2000) New trends in quantum structures. Kluwer, Dordrecht; Ister Science, Bratislava
Dvurečenskij A, Vetterlein T (2001a) Congruences and states on pseudoeffect algebras. Found Phys Lett 14(5):425–446
Dvurečenskij A, Vetterlein T (2001b) Pseudoeffect algebras: I. Basic properties. Int J Theor Phys 40:685–701
Dvurečenskij A, Vetterlein T (2001c) Pseudoeffect algebras: II. Group representations. Int J Theor Phys 40:703–726
Foulis D, Bennett MK (1994) Effect algebra and unsharp quantum logics. Found Phys 24:1331–1352
Georgescu G, Iorgulescu A (2001) Pseudo-MV algebra. Mult Val Logic 6:95–135
Giuntini R, Greuling H (1989) a formal language for unsharp properties. Found Phys 19:931–945
Gudder S, Pulmannová S (1997) Quotient of partial Abelian monoids. Algebra Universalis 38:395–421
Jenča G (2001) A note on ideals in generalized effect algebra. Soft Comput 5:376–380
Jenča G, Pulmannová S (1999) Ideals and quotients in lattice ordered effect algebra. Tatra Mt Math Publ 16:81–85
Kôpka F (1992) D-posets of fuzzy sets. Tatra Mt Math Publ 1:83–87
Li H-Y, Li S-G (2008) Congruences and ideals in pseudoeffect algebra. Soft Comput 12:487–492
Pulmannová S (1997) Congruences in partial Abelian monoids. Algebra Universalis 37:119–140
Acknowledgments
This work was supported by the Natural Science Foundation of China (11026201), the Foundation of Shaanxi Province (2009JK452) and the Foundation of XPU (09XG10).
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Li, HY., Peng, JG. Quotients and weakly algebraic sets in pseudoeffect algebras. Soft Comput 16, 485–492 (2012). https://doi.org/10.1007/s00500-011-0750-z
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DOI: https://doi.org/10.1007/s00500-011-0750-z