Article
Full entry |
PDF
(1.2 MB)
Feedback
PDF
(1.2 MB)
Feedback
Keywords:
domination; Frank t-norm; Hamacher $t$-norm
domination; Frank t-norm; Hamacher $t$-norm
Summary:
Domination is a relation between general operations defined on a poset. The old open problem is whether domination is transitive on the set of all t-norms. In this paper we contribute partially by inspection of domination in the family of Frank and Hamacher t-norms. We show that between two different t-norms from the same family, the domination occurs iff at least one of the t-norms involved is a maximal or minimal member of the family. The immediate consequence of this observation is the transitivity of domination on both inspected families of t-norms.
References:
[1] Bodenhofer U.: A Similarity-Based Generalization of Fuzzy Orderings. (Schriftenreihe der Johannes–Kepler–Universität Linz, Volume C 26.) Universitätsverlag Rudolf Trauner, Linz 1999 Zbl 1113.03333
[2] Baets B. De, Mesiar R.: Pseudo-metrics and $T$-equivalences. J. Fuzzy Math. 5 (1997), 471–481 MR 1457163
[3] Baets B. De, Mesiar R.: $T$-partitions. Fuzzy Sets and Systems 97 (1998), 211–223 DOI 10.1016/S0165-0114(96)00331-4 | MR 1645614 | Zbl 0930.03070
[4] Drewniak J., Drygaś, P., Dudziak U.: Relation of domination. In: FSTA 2004 Abstracts, pp. 43–44
[5] Frank M. J.: On the simultaneous associativity of $F(x,y)$ and $x+y-F(x,y)$. Aequationes Math. 19 (1979), 194–226 DOI 10.1007/BF02189866 | MR 0556722 | Zbl 0444.39003
[6] Hamacher H.: Über logische Verknüpfungen unscharfer Aussagen und deren zugehörige Bewertungsfunktionen. Progress in Cybernetics and Systems Research, Hemisphere Publ. Comp., New York 1975, pp. 276–287
[7] Hamacher H.: Über logische Aggregationen nicht-binär explizierter Entscheidungskriterien. Rita G. Fischer Verlag, Frankfurt 1978
[8] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000 MR 1790096 | Zbl 1087.20041
[9] Saminger S., Mesiar, R., Bodenhofer U.: Domination of aggregation operators and preservation of transitivity. Internat. J. Uncertain. Fuzziness Knowledge-based Systems 10 (2002), 11–35 DOI 10.1142/S0218488502001806 | MR 1962666 | Zbl 1053.03514
[10] Saminger S.: Aggregation in Evaluation of Computer-assisted Assessment. (Schriftenreihe der Johannes–Kepler–Universität Linz, Volume C 44.) Universitätsverlag Rudolf Trauner, Linz 2005 Zbl 1067.68143
[11] Sherwood H.: Characterizing dominates on a family of triangular norms. Aequationes Math. 27 (1984), 255–273 DOI 10.1007/BF02192676 | MR 0762685 | Zbl 0598.26032
[12] Schweizer B., Sklar A.: Probabilistic Metric Spaces. North–Holland, New York 1983 MR 0790314 | Zbl 0546.60010
[13] Valverde L.: On the structure of F-indistinguishability operators. Fuzzy Sets and Systems 17 (1985), 313–328 DOI 10.1016/0165-0114(85)90096-X | MR 0819367 | Zbl 0609.04002
Search
Partner of
