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Pseudo-BL algebras and pseudo-effect algebras. (English) Zbl 1176.03047
Summary: Pseudo-BL algebras and pseudo-effect algebras arose in two rather different fields: fuzzy logics and quantum logics. In this paper, by introducing the notion of pseudo-weak MV-effect algebras, which is a noncommutative generalization of the notion of weak MV-effect algebra, we investigate the mutual relationship between pseudo-BL algebras and pseudo-effect algebras. We prove that a dual pseudo-BL algebra can be restricted under a certain condition to a pseudo-weak MV-effect algebra, and a pseudo-weak MV-effect algebra can be extended to a dual pseudo-BL algebra under a certain condition. Moreover, we give some examples of pseudo-BL algebras which correspond to some pseudo-weak MV-effect algebras. Finally, we establish the relationship between pseudo-MV algebras and pseudo-MV effect algebras.

MSC:
03G25 Other algebras related to logic
03B52 Fuzzy logic; logic of vagueness
03G12 Quantum logic
06D35 MV-algebras
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[1] Di Nola, A.; Georgescu, G.; Iorgulescu, A., Pseudo-BL algebras I, II, multiple-valued logic, 8, 673-714, 717-750, (2002) · Zbl 1028.06008
[2] Dvurec˘nskij, A.; Vetterlein, T., Pseudo-effect algebras: I. basic properties, Internat. J. theoret. phys., 40, 685-701, (2001) · Zbl 0994.81008
[3] Dvurec˘enskij, A.; Vetterlein, T., Non-commutative algebras and quantum structures, Internat. J. theoret. phys., 43, 7-8, 1599-1612, (2004) · Zbl 1075.81010
[4] Flondor, P.; Georgescu, G.; Iorgulescu, A., Pseudo-t-norms and pseudo-BL algebras, Soft comput., 5, 355-371, (2001) · Zbl 0995.03048
[5] Foulis, D.J.; Bennett, M.K., Effect algebras and unsharp quantum logics, Found. phys., 24, 1325-1346, (1994) · Zbl 1213.06004
[6] Georgescu, G.; Iorgulescu, A., Pseudo-MV algebras, Multiple-valued logic, 6, 95-135, (2001) · Zbl 1014.06008
[7] Georgescu, G.; Leustean, L., Some classes of pseudo-BL algebras, J. austral. math. soc., 73, 127-153, (2002) · Zbl 1016.03069
[8] Georgescu, G.; Popescu, A., Non-commutative fuzzy structures and pairs of weak negations, Fuzzy sets and systems, 143, 129-155, (2004) · Zbl 1036.06007
[9] Hájek, P., Observations on non-commutative fuzzy logic, Soft comput., 8, 38-43, (2003) · Zbl 1075.03009
[10] Hájek, P., Fuzzy logics with non-commutative conjunction, J. logic comput., 13, 469-479, (2003) · Zbl 1036.03018
[11] A. Iorgulescu, Classes of pseudo-BCK algebras-Part I, Part II, J. Multiple-Valued Logic Soft Comput. 12(1-2) (2006), 71-130; 12(5-6): 575-629.
[12] Jenei, S.; Montagna, F., A proof of standard completeness for non-commutative monoidal t-norm logic, Neural network world, 5, 481-489, (2003)
[13] Leustean, I., Non-commutative lukasiewicz propositional logic, Arch. math. logic, 45, 191-213, (2006) · Zbl 1096.03020
[14] Rachunek, J., Prime spectra of non-commutative generalizations of MV-algebras, Algebra universalis, 48, 151-169, (2002) · Zbl 1058.06015
[15] Vetterlein, T., BL-algebras and quantum structures, Math. slovaca, 54, 2, 127-141, (2004) · Zbl 1065.03049
[16] Vetterlein, T., BL-algebras and effect algebras, Soft comput., 9, 8, 557-564, (2005) · Zbl 1094.03059
[17] Yetter, D.N., Quantales and (non-commutative) linear logic, J. symbolic logic, 55, 1, 41-64, (1990) · Zbl 0701.03026
[18] X.H. Zhang, On non-commutative fuzzy logic structures, Proc. Fifth Int. Conf. Mach. Learning and Cybernetics, IEEE 06EX1263, Vol. 3(7), 2006, pp. 1867-1871.
[19] Zhang, X.H., Non-commutative fuzzy logic system \(\textit{PL}^*\) and its completeness, Acta math. sinica, 50, 2, 421-442, (2007), (Chinese Series) · Zbl 1121.03302
[20] Zhang, X.H.; Li, W.H., On pseudo-BL algebras and BCC-algebras, Soft comput., 10, 941-952, (2006) · Zbl 1112.03057
[21] Zhou, X.N.; Li, Q.G.; Wang, G.J., Residuated lattices and lattice effect algebras, Fuzzy sets and systems, 158, 904-914, (2007) · Zbl 1122.81016
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