Abstract
Congruences and ideals in pseudo-effect algebras and their total algebra versions are studied. It is shown that every congruence of the total algebra induces a Riesz congruence in the corresponding pseudo-effect algebra. Conversely, to every normal Riesz ideal in a pseudo-effect algebra there is a total algebra, in which the given ideal induces a congruence of the total algebra. Ideals of total algebras corresponding to lattice-ordered pseudo-effect algebras are characterized, and it is shown that they coincide with normal Riesz ideals in the pseudo-effect algebras.
Similar content being viewed by others
References
Avallone A, Vitolo P (2003) Congruences and ideals of effect algebras. Order 20:67–77
Beltrametti E, Cassinelli G (1981) The logic of quantum mechanics. Addison-Wesley, Reading
Butnariu D, Klement P (1993) Triangular norm-based measures and games with fuzzy coalitions. Kluwer, Dordrecht
Chajda I, Kühr J (2009) Pseudo-effect algebras as total algebras. Int J Theor Phys (to appear)
Chajda I, Halaš R, Kühr J (2009a) Many-valued quantum algebras. Algebra Univers 63:63–90
Chajda I, Halaš R, Kühr J (2009b) Every effect algebra can be made into a total algebra. Algebra Univers 61:139–150
Chang CC (1958) Algebraic analysis of many valued logics. Trans Am Math Soc 88:467–490
Chevalier G, Pulmannová S (2000) Some ideal lattices in partial Abelian monoids and effect algebras. Order 17:72–92
Cignoli R, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning. Kluwer, Dordrecht
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) Congruences and states on pseudo-effect algebras. Found Phys Lett 14:422–446
Foulis D, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1331–1352
Giuntini R (1996) Quantum MV-algebras. Studia Logica 56:303–417
Giuntini R, Grueling R (1989) Toward a formal language for unsharp properties. Found Phys 19:931–945
Giuntini R, Pulmannová S (2000) Ideals and congruences in effect algebras and QMV-algebras. Comm Algebra 28:1567–1592
Georgescu G, Iorgulescu A (2001) Pseudo-MV algebras. Multi-Valued Logic 6:95–135
Gudder S (1995) Total extensions of effect algebras. Found Phys Lett 8:243–252
Gudder S, Pulmannová S (1997) Quotients of partial abelian monoids. Algebra Univers 38:395–421
Jenča G (2000) Notes on R 1-ideals in partal Abelian monoids. Algebra Univers 43:307–319
Jenča G, Pulmannová S (2001) Ideals and quotients in lattice ordered effect albebras. Soft Comput 5:376–380
Ježek J, Quackenbush R (1990) Directoids: algebraic models of up-directed sets. Algebra Univers 27:49–69
Kôpka F, Chovanec F (1994) D-posets. Math Slovaca 44:21–34
Li H-Y, Li S-G (2008) Congruences and ideals in pseudo-effect algebras. Soft Comput 12:487–492
Ludwig G (1983) Foundations of quantum mechanics. Springer, New York
Pták P, Pulmannová S (1991) Orthomodular structures as quantum logics. Kluwer, Dordrecht
Pulmannová S (1997) Congruences in partial abelian semigroups. Algebra Univers 37:119–140, Corrigendum ibid. 42:59–60
Pulmannová S (2003) Generalized Sasaki projections and Riesz ideals in pseudoeffect algebras. Int J Theor Phys 42:1413–1423
Pulmannová S, Vinceková E (2009) Congruences and ideals in lattice effect algebras as basic algebras. Kybernetika (to appear)
Rachůnek J (2002) A non-commutative generalization of MV-algebras. Czech J Math 52:255–273
Acknowledgments
This work was supported by the Slovak Research and Development Agency under the contract No. LPP-0199-07 and APVV-0071-06; by the Slovak-Italian project SK-IT 0016-08, ERDF OP R&D Project CE QUTE ITMS 26240120009 and grant VEGA 2/0032/09.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pulmannová, S., Vinceková, E. Congruences and ideals in pseudo effect algebras as total algebras. Soft Comput 14, 1209–1215 (2010). https://doi.org/10.1007/s00500-009-0532-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-009-0532-z