Triangular norms. Position paper III: Continuous t-norms

Fuzzy Sets and Systems

Volumn 145, Issue 3, 2004, Pages 439-454

  Klement, Erich Peter ^a   Mesiar, Radko ^b,c   Pap, Endre ^d  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

^b SLOVAK UNIVERSITY OF TECHNOLOGY   (Slovakia)

^c INSTITUTE OF INFORMATION THEORY AND AUTOMATION   (Czech Republic)

^d UNIVERSITY OF NOVI SAD   (Serbia)

Author keywords

Additive generator; Continuous triangular norm; Ordinal sum

Indexed keywords

ALGEBRA; FORMAL LOGIC; FUNCTIONS; MATHEMATICAL OPERATORS; PROBABILITY DISTRIBUTIONS; SET THEORY;

ADDITIVE GENERATORS; AXIOMS; CONTINUOUS TRIANGULAR NORM; ORDINAL SUM;

FUZZY SETS;

EID: 2942666059     PISSN: 01650114     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0165-0114(03)00304-X     Document Type: Article

References (40)

1. Aczél J. Sur les opérations definies pour des nombres réels. Bull. Soc. Math. France. 76:1949;59-64.
2. Aczél J. Lectures on Functional Equations and their Applications. 1966;Academic Press, New York.
3. Aczél J., Alsina C. Characterizations of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgements. Methods Oper. Res. 48:1984;3-22.
4. Alsina C. On a method of Pi-Calleja for describing additive generators of associative functions. Aequationes Math. 43:1992;14-20.
5. Alsina C., Gimenez J. Sobre. L -órdenes entre t-normas estrictas Stochastica. 8:1984;85-89.
6. Bacchelli B. Representation of continuous associative functions. Stochastica. 10:1986;13-28.
7. Butnariu D., Klement E.P. Triangular Norm-Based Measures and Games with Fuzzy Coalitions. 1993;Kluwer Academic Publishers, Dordrecht.
8. Clifford A.H. Naturally totally ordered commutative semigroups. Amer. J. Math. 76:1954;631-646.
9. Clifford A.H., Preston G.B. The Algebraic Theory of Semigroups. 1961;American Mathematical Society, Providence, RI.
10. Climescu A.C. Sur l'équation fonctionelle de l'associativité Bull. École Polytechn. Iassy. 1:1946;1-16.
11. Craigen R., Páles Z. The associativity equation revisited. Aequationes Math. 37:1989;306-312.
12. Dombi J. A general class of fuzzy operators, the De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets and Systems. 8:1982;149-163.
13. Frank M.J. On the simultaneous associativity of. F(x,y) and x+y-F(x,y) Aequationes Math. 19:1979;194-226.
14. Hájek P. Observations on the monoidal t-norm logic. Fuzzy Sets and Systems. 132:2002;107-112.
15. Hamacher H. Über logische Aggregationen nicht-binär explizierter Entscheidungskriterien. 1978;Rita G. Fischer Verlag, Frankfurt.
16. Hardy G.H., Littlewood J.E., Pólya G. Inequalities. 1952;Cambridge University Press, Cambridge.
17. K.H. Hofmann, J.D. Lawson, Linearly ordered semigroups: historic origins and A.H. Clifford's influence, in: K.H. Hofmann, M.W. Mislove (Eds.), Semigroup Theory and its Applications, Vol. 231, London Mathematical Society Lecture Note, Cambridge University Press, Cambridge, 1996, pp. 15-39.
18. Jenei S. On Archimedean triangular norms. Fuzzy Sets and Systems. 99:1998;179-186.
19. Jenei S., Fodor J.C. On continuous triangular norms. Fuzzy Sets and Systems. 100:1998;273-282.
20. Kimberling C. On a class of associative functions. Publ. Math. Debrecen. 20:1973;21-39.
21. Klein-Barmen F. Über gewisse Halbverbände und kommutative Semigruppen II. Math. Z. 48:1942-43;715-734.
22. Klement E.P., Mesiar R., Pap E. A characterization of the ordering of continuous t-norms. Fuzzy Sets and Systems. 86:1997;189-195.
23. Klement E.P., Mesiar R., Pap E. Triangular Norms. 2000;Kluwer Academic Publishers, Dordrecht.
24. E.P. Klement, R. Mesiar, E. Pap, Triangular norms, Position paper I: basic analytical and algebraic properties, Fuzzy Sets and Systems (in press).
25. E.P. Klement, R. Mesiar, E. Pap, Triangular norms. Position paper II: general constructions and parameterized families, submitted for publication.
26. Kolesárová A. A note on Archimedean triangular norms. BUSEFAL. 80:1999;57-60.
27. G.M. Krause, A strengthened form of Ling's theorem on associative functions, Ph.D. Thesis, Illinois Institute of Technology, Chicago, 1981.
28. Ling C.M. Representation of associative functions. Publ. Math. Debrecen. 12:1965;189-212.
29. Mesiar R., Navara M. Diagonals of continuous triangular norms. Fuzzy Sets and Systems. 104:1999;35-41.
30. Mostert P.S., Shields A.L. On the structure of semi-groups on a compact manifold with boundary. Ann. Math. (2). 65:1957;117-143.
31. T semigroups of probability distribution functions II Aequationes Math. 17:1978;19-40.
32. Nguyen H.T., Kreinovich V., Wojciechowski P. Strict Archimedean t-norms and t-conorms as universal approximators. Internat. J. Approx. Reason. 18:1998;239-249.
33. P. Pi-Calleja, Las ecuaciones funcionales de la teoria de magnitudes, in: Proc. Segundo Symp. de Matemática, Villavicencio, Mendoza, Coni, Buneos Aires, 1954, pp. 199-280.
34. Preston G.B. A.H. Clifford. an appreciation of his work on the occasion of his 65th birthday Semigroup Forum. 7:1974;32-57.
35. Schweizer B., Sklar A. Associative functions and statistical triangle inequalities. Publ. Math. Debrecen. 8:1961;169-186.
36. Schweizer B., Sklar A. Associative functions and abstract semigroups. Publ. Math. Debrecen. 10:1963;69-81.
37. Schweizer B., Sklar A. Probabilistic Metric Spaces. 1983;North-Holland, New York.
38. Viceník P. Additive generators of non-continuous triangular norms. Rodabaugh S.E., Klement E.P. Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets. 2003;441-454 Kluwer Academic Publishers, Dordrecht. (Chapter 18).
39. Weber S. A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms. Fuzzy Sets and Systems. 11:1983;115-134.
40. Yager R.R. On a general class of fuzzy connectives. Fuzzy Sets and Systems. 4:1980;235-242.