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Cancellativity properties for t-norms and t-subnorms. (English) Zbl 1162.03013

The authors establish that the zooms corresponding to non-trivial Archimedean classes of a t-subnorm are Archimedean t-subnorms. Further, they characterize those t-subnorms whose Archimedean components are cancellative, conditionally cancellative, weakly cancellative, or weakly conditionally cancellative, and the relationships between these types of cancellativity are established. An open problem of E. P. Klement, R. Mesiar and E. Pap [Fuzzy Sets Syst. 145, No. 3, 471–479 (2004; Zbl 1050.03019)] is also solved negatively.

MSC:

03B52 Fuzzy logic; logic of vagueness

Citations:

Zbl 1050.03019
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References:

[1] Budinčevič, M.; Kurilič, M. S., A family of strict and discontinuous triangular norms, Fuzzy Sets and Systems, 95, 381-384 (1998) · Zbl 0922.04006
[2] Esteva, F.; Godo, L., Monoidal t-norm based fuzzy logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems, 124, 271-288 (2001) · Zbl 0994.03017
[3] Fodor, J., Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems, 69, 141-156 (1995) · Zbl 0845.03007
[4] Fuchs, L., Partially Ordered Algebraic Systems (1963), Pergamon Press: Pergamon Press Oxford · Zbl 0137.02001
[5] Hájek, P., Observations on the monoidal t-norm logic, Fuzzy Sets and Systems, 132, 107-112 (2002) · Zbl 1012.03035
[6] Hewitt, E.; Zuckerman, H. S., The \(l_1\)-algebra of a commutative semigroup, Transactions of the American Mathematical Society, 83, 70-97 (1956) · Zbl 0072.12701
[7] Iliadis, L. S.; Spartalis, S.; Tachos, S., Application of fuzzy t-norms towards a new Artificial Neural Networks’ evaluation framework: A case from wood industry, Information Sciences, 178, 20, 3828-3839 (2008)
[8] Jenei, S., A note on the ordinal sum theorem and its consequence for the construction of triangular norms, Fuzzy Sets and Systems, 126, 199-205 (2002) · Zbl 0996.03508
[9] Jenei, S.; Montagna, F., A general method for constructing left-continuous t-norms, Fuzzy Sets and Systems, 136, 263-282 (2003) · Zbl 1020.03020
[10] Klement, E. P.; Mesiar, R.; Pap, E., Triangular norms, Trends in Logic, vol. 8 (2000), Kluwer Academic Publishers · Zbl 0972.03002
[11] Klement, E. P.; Mesiar, R.; Pap, E., Problems on triangular norms and related operators, Fuzzy Sets and Systems, 145, 471-479 (2004) · Zbl 1050.03019
[12] Klement, E. P.; Mesiar, R.; Pap, E., Archimedean components of triangular norms, Journal of the Australian Mathematical Society, 78, 239-255 (2005) · Zbl 1087.20041
[13] Krasilnikova, Y. I.; Novikov, B. V., On quasi-separative semigroups, Semigroup Forum, 70, 347-355 (2005) · Zbl 1095.20036
[14] D. Kyselova, Aggregation operators-based multicriteria decision making, PhD Thesis, STU, Bratislava, 2007.; D. Kyselova, Aggregation operators-based multicriteria decision making, PhD Thesis, STU, Bratislava, 2007.
[15] Maes, K. C.; De Baets, B., Isomorphic continuous connectives in fuzzy logic, Journal of Intelligent and Fuzzy Systems, 16, 4, 273-279 (2005)
[16] K.C. Maes, B. De Baets, Rotation-invariant t-norms: where triple rotation and rotation-annihilation meet, in press, doi:10.1016/j.fss.2008.09.020.; K.C. Maes, B. De Baets, Rotation-invariant t-norms: where triple rotation and rotation-annihilation meet, in press, doi:10.1016/j.fss.2008.09.020. · Zbl 1182.03052
[17] A. Mesiarová, Special classes of triangular norms, PhD Thesis, Mathematical Institute of SAS, Bratislava, 2005.; A. Mesiarová, Special classes of triangular norms, PhD Thesis, Mathematical Institute of SAS, Bratislava, 2005.
[18] Mesiarová, A., H-transformation of t-norms, Information Sciences, 176, 1531-1545 (2006) · Zbl 1094.03040
[19] Montagna, F.; Noguera, C.; Horčík, R., On weakly cancellative fuzzy logics, Journal of Logic and Computation, 16, 423-450 (2006) · Zbl 1113.03021
[20] Nobuhara, H.; Pedrycz, W.; Sessa, S.; Hirota, K., A motion compression/reconstruction method based on max t-norm composite fuzzy relational equations, Information Sciences, 176, 17, 2526-2552 (2006) · Zbl 1102.68698
[21] Oussalah, M., On the use of Hamacher’s t-norms family for information aggregation, Information Sciences, 153, 107-154 (2003) · Zbl 1069.68611
[22] Ouyang, Y.; Fang, J.; Li, J., A conditionally cancellative left-continuous t-norm is not necessarily continuous, Fuzzy Sets and Systems, 157, 2328-2332 (2006) · Zbl 1115.03015
[23] Schwarz, Š., Semigroups satisfying some weakened forms of the cancellation law (in Slovak), Mat.-Fyz. Časopis Slovensk. Akad. Vied., 6, 149-158 (1956)
[24] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), Elsevier Science: Elsevier Science New York · Zbl 0546.60010
[25] Urbański, M.; Wa¸sowski, J., Fuzzy arithmetic based on boundary weak t-norms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 13, 27-37 (2005) · Zbl 1069.03052
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