Abstract
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.
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References
Avallone A, Vitolo P (2003) Congruences and ideals of effect algebras. Order 20:67–77
Dvurečenskij A (2003) Ideals of pseudo effect algebras and their applications. Tatra Mt Math Publ 27:45–65
Dvurečenskij A (2013) Kite pseudo effect algebras. Found Phys 43:1314–1338
Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer, Dordrecht
Dvurečenskij A, Vetterlein T (2001a) Pseudo effect algebras I. Basic properties. Int J Theor Phys 40:685–701
Dvurečenskij A, Vetterlein T (2001b) Pseudo effect algebras II. Group representation. Int J Theor Phys 40:703–726
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 89–111
Dvurečenskij A, Vetterlein T (2001d) Congruences and states on pseudo effect algebras. Found Phys Lett 14:425–446
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 89–111
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
Foulis DJ, Pulmannová S (2015) Unitizing a generalized pseudo effect algebra. Order 32:311–332
Foulis DJ, Pulmnnová S, Vinceková E (2013) The exocenter and type decompositions of a generalized pseudoeffect algebra. Discuss Math Gen Algebra Appl 33:13–47
Foulis DJ, Pulmannová S, Vinceková E (2016) Unitizations of generalized pseudoeffect algebras and their ideals. Order 33:311–332
Gudder S, Pulmannová S (1997) Quotients of partial abelian monoids. Algebra Univ 38:395–421
Haiyang Li, Shenggang Li (2008) Congruences and ideals in pseudo effect algebras. Soft Comput 12:487–492
Jenča G (2000) Subcentral ideals in generalized effect algebras. Int J Theor Phys 19:745–755
Pulmannová S, Vinceková E (2007) Riesz ideals in generalized effect algebras and in their unitizations. Algebra Univ 57:393–417
Riečanová Z, Marinová I (2005) Generalized homogeneous, prelattice and MV-effect algebras. Kybernetika 41:129–142
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 D (2011) Weak commutative pseudo effect algebras. Int J Theor Phys 50:1186–1197
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The author was supported by Research and Development Support Agency under the Contract APVV-16-0073 and Grant VEGA 2/0069/16.
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Communicated by A. Di Nola.
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Pulmannová, S. Congruences and ideals in generalized pseudoeffect algebras revisited. Soft Comput 23, 735–745 (2019). https://doi.org/10.1007/s00500-018-3107-z
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DOI: https://doi.org/10.1007/s00500-018-3107-z