×

Different types of continuity of triangular norms revisited. (English) Zbl 1081.26024

The authors study many continuity conditions for t-norms \(T\), namely (in the order of decreasing generality): – continuity at \((1,1)\), – broder continuity (continuity at all points of \(\{1\}\times[0,1]\)), – left continuity, – continuity, – \(k\)-Lipschitz property for \(k>1\), – \(1\)-Lipschitz property, – \(\| \cdot\| _p\)-stability (\(| T(x_1,y_1)-T(x_2,y_2)| \leq(| x_1-x_2| ^p+| y_1-y_2| ^p)^{1/p}\)), \(p>1\). Numerous results on these types of continuity are collected, in particular their relations to ordinal sums and additive generators. Particular attention is paid to the correspondence with the properties of diagonal sections, i.e., the functions \(x\mapsto T(x,x)\). Also Schur concavity, \[ T(x,y)\leq T(\lambda x+(1-\lambda)y, (1-\lambda)x+\lambda y) \] for all \(\lambda\in[0,1]\) and its special form for \(\lambda=1/2\), \[ T(x,y)\leq T((x+y)/2,(x+y)/2), \] are put into this context. It is a useful and complete survey of contemporary knowledge in this field, accompanied by numerous interesting examples.

MSC:

26E50 Fuzzy real analysis
03E72 Theory of fuzzy sets, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1007/s00010-003-2673-y · Zbl 1077.39021 · doi:10.1007/s00010-003-2673-y
[2] DOI: 10.1007/978-94-017-3602-2 · doi:10.1007/978-94-017-3602-2
[3] DOI: 10.2307/2372706 · Zbl 0055.01503 · doi:10.2307/2372706
[4] DOI: 10.1090/surv/007.1 · doi:10.1090/surv/007.1
[5] Climescu A. C., Bull. École Polytechn. Iassy 1 pp 1–
[6] Darsow W. F., Publ. Math. Debrecen 31 pp 253–
[7] DOI: 10.1007/978-1-4615-4429-6_11 · doi:10.1007/978-1-4615-4429-6_11
[8] DOI: 10.1016/j.fss.2004.11.014 · Zbl 1065.03035 · doi:10.1016/j.fss.2004.11.014
[9] Durante F., Int. Math. J. 3 pp 893–
[10] DOI: 10.1016/0165-0114(94)00210-X · Zbl 0845.03007 · doi:10.1016/0165-0114(94)00210-X
[11] DOI: 10.1007/978-94-017-1648-2 · doi:10.1007/978-94-017-1648-2
[12] DOI: 10.1007/BF02189866 · Zbl 0444.39003 · doi:10.1007/BF02189866
[13] DOI: 10.1016/j.fss.2004.05.005 · Zbl 1074.03023 · doi:10.1016/j.fss.2004.05.005
[14] Gottwald S., A Treatise on Many-Valued Logic (2001) · Zbl 1048.03002
[15] DOI: 10.1007/978-94-015-8449-4 · doi:10.1007/978-94-015-8449-4
[16] DOI: 10.1007/978-94-011-5300-3 · doi:10.1007/978-94-011-5300-3
[17] DOI: 10.1017/CBO9780511661877.004 · doi:10.1017/CBO9780511661877.004
[18] DOI: 10.1007/BF01180038 · Zbl 0028.00601 · doi:10.1007/BF01180038
[19] DOI: 10.1016/0022-247X(82)90015-4 · Zbl 0491.28003 · doi:10.1016/0022-247X(82)90015-4
[20] Klement E. P., Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms (2005) · Zbl 1063.03003
[21] DOI: 10.1007/978-94-015-9540-7 · doi:10.1007/978-94-015-9540-7
[22] Klement E. P., Illinois J. Math. 45 pp 1393–
[23] DOI: 10.1016/S0165-0114(03)00303-8 · Zbl 1050.03019 · doi:10.1016/S0165-0114(03)00303-8
[24] DOI: 10.1016/j.fss.2003.06.007 · Zbl 1038.03027 · doi:10.1016/j.fss.2003.06.007
[25] DOI: 10.1016/S0165-0114(03)00327-0 · Zbl 1059.03012 · doi:10.1016/S0165-0114(03)00327-0
[26] DOI: 10.1016/S0165-0114(03)00304-X · Zbl 1059.03013 · doi:10.1016/S0165-0114(03)00304-X
[27] DOI: 10.1016/0165-0114(91)90166-N · Zbl 0733.28012 · doi:10.1016/0165-0114(91)90166-N
[28] Klir G. J., Fuzzy Sets, Uncertainty, and Information (1988) · Zbl 0675.94025
[29] Ling C. M., Publ. Math. Debrecen 12 pp 189–
[30] Menger K., Proc. Natl. Acad. Sci. USA 8 pp 535–
[31] DOI: 10.1016/S0165-0114(98)00256-5 · Zbl 0972.03052 · doi:10.1016/S0165-0114(98)00256-5
[32] Mesiarová A., J. Electrical Eng. 52 pp 7–
[33] DOI: 10.2307/1969668 · Zbl 0096.01203 · doi:10.2307/1969668
[34] DOI: 10.1007/BF01818536 · Zbl 0386.22005 · doi:10.1007/BF01818536
[35] DOI: 10.1007/3-540-48236-9 · doi:10.1007/3-540-48236-9
[36] DOI: 10.1016/0165-0114(90)90207-M · Zbl 0722.28015 · doi:10.1016/0165-0114(90)90207-M
[37] Pap E., Atti Sem. Mat. Fis. Univ. Modena 39 pp 345–
[38] Pap E., Null-Additive Set Functions (1995) · Zbl 0856.28001
[39] DOI: 10.1007/BF02315965 · Zbl 0275.01014 · doi:10.1007/BF02315965
[40] Schwarz Š., Mat.-Fyz. Časopis Slovensk. Akad. Vied 6 pp 149–
[41] DOI: 10.2140/pjm.1960.10.313 · Zbl 0091.29801 · doi:10.2140/pjm.1960.10.313
[42] Schweizer B., Publ. Math. Debrecen 10 pp 69–
[43] Schweizer B., Probabilistic Metric Spaces (1983) · Zbl 0546.60010
[44] Šerstnev A. N., Kazan. Gos. Univ. Učen. Zap. 122 pp 3–
[45] Timan A. F., International Series of Monographs in Pure and Applied Mathematics 34, in: Theory of approximation of functions of a real variable (1963)
[46] Tkadlec J., Tatra Mt. Math. Publ. 16 pp 187–
[47] DOI: 10.1016/0022-247X(84)90061-1 · Zbl 0614.28019 · doi:10.1016/0022-247X(84)90061-1
[48] DOI: 10.1016/0165-0114(80)90013-5 · Zbl 0443.04008 · doi:10.1016/0165-0114(80)90013-5
[49] DOI: 10.1016/S0165-0114(98)00353-4 · Zbl 0931.68117 · doi:10.1016/S0165-0114(98)00353-4
[50] DOI: 10.1016/S0019-9958(65)90241-X · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.