Abstract
We prove that there is an order isomorphism between the lattice of all normal Riesz ideals and the lattice of all Riesz congruences in upwards directed generalized pseudoeffect algebras (or GPEAs, for short). We give a sufficient and necessary condition under which a normal Riesz ideal I of a weak commutative generalized pseudoeffect algebra P is a normal Riesz ideal also in the unitization \(\widehat{P}\) of P. These results extend those obtained recently by Avalllone, Vitolo, Pulmannová and Vinceková for effect algebras. At the same time, we give the conditions under which the quotient of a generalized pseudoeffect algebra P is a generalized effect algebra and linearly ordered generalized pseudoeffect algebra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
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 17:75–92
Dvurečenskij A (2002) Pseudo MV-algebras are intervals in ℓ-groups. J Austral Math Soc 70:427–445
Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer, Dordrecht
Dvurečenskij A, Vetterlein T (2001a) Pseudoeffect algebras. I. Basic properties. Inter J Theor Phys 40:685–701
Dvurečenskij A, Vetterlein T (2001b) Pseudeffect algebras. II. Group representations. Inter J Theor Phys 40:703–726
Dvurečenskij A, Vetterlein T (2001c) Congruences and states on pseudoeffect algebras. Found Phys Lett 14:425–446
Dvurečenskij A, Vetterlein T (2001d) Generalized pseudo-effect algebras. Lectures on soft computing and fuzzy logic. Adv Soft Comput 89–111
Dvurečenskij A, Vetterlein T (2002) Algebras in the positive cone of po-groups. Order 19:127–146
Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1325–1346
Georgescu G, Iorgulescu A (2001) Pseudo-MV algebras. Multi Val Logic 6:95–135
Gudder S, Pulmannová S (1997) Quotients of partial abelian monoids. Algebra Universalis 38:395–421
Haiyang L, Shenggang L (2008) Congruences and ideals in pseudoeffect algebras. Soft Comput 12:487–492
Hájek P (2003) Observations on non-commutative fuzzy logic. Soft Comput 8:38–43
Hedlíková J, Pulmannová S (1996) Generalized difference posets and orthoalgebras. Acta Math Univ Comenianae 45:247–279
Kalmbach G (1983) Orthomodular Lattices. London Math. Soc. Monographs, vol 18. Academic Press, London
Kôpka F, Chovanec F (1994) D-posets. Math Slovaca 44:21–34
Pulmannová S, Vinceková E (2007) Riesz ideals in generalized effect algebras and in their unitizations. Algebra Universalis 57:393–417
Rachunek J (2002) A non-commutative generalization of MV-algebras. Czechoslovak Math J 52:255–273
Shang Y (2005) Studies on effect algebras and pseudoeffect algebras in quantum logics. PhD Thesis, Shaanxi Normal University (in Chinese)
Xie Y, Li Y (manuscript) Weak commutative pseudoeffect algebras
Acknowledgements
The authors would like to thank the referees for their valuable suggestions which improved the readability of the paper. This work is supported by National Science Foundation of China (Grant No. 60873119, 60773224) and the Research Foundation for the Doctorial Program of Higher School of Ministry of Education (Grant No. 200807180005).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xie, Y., Li, Y. Riesz ideals in generalized pseudo effect algebras and in their unitizations. Soft Comput 14, 387–398 (2010). https://doi.org/10.1007/s00500-009-0412-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-009-0412-6