In this paper, we present some new results in the form of inequalities connecting transitivity of the given fuzzy preference relation with its consistent behaviour. The additive and multiplicative generators of t-norms and t-conorms play the key role in establishing these results.
M.Baczyński and B.Jayaram, Fuzzy Implications, Springer Publisher, Berlin Heidelberg (2008).
2.
M.Baczyński, P.Grzegorzewski, R.Mesiar, P.Helbin and W.Niemyska, Fuzzy implications based on semicopulas, Fuzzy Sets and Systems323 (2017), 138–151.
3.
I.Beg and S.Ashraf, Fuzzy equivalence relations, Kuwait J, Science and Engineering35(1A) (2008), 33–51.
4.
J.C.Bezdek and J.D.Harris, Fuzzy partitions and relations: An axiomatic basis for clustering, Fuzzy Sets and Systems1 (1978), 111–127.
5.
D.Boixader, J.Jacas and J.Recasens, Fuzzy equivalence relations: Advanced material. In: Fundamentals of fuzzy sets(Chapter 5 D. Dubois, & H. Prade).Boston: Kluwer Academic Publishers (2000).
6.
D.Boixader and J.Recasens, Discretization of fuzzy transitive relations, IEEE international conference on fuzzy systems, Taipei, Taiwan (2011).
7.
B.De Beats and E.E.Kerre, Fuzzy relations and applications, Advances in Electronics and Electron Physics89 (1994), 255–324.
8.
W.Caiping and L.Wang, Some results on the relationships between transitivity-related indicators of fuzzy relations, International Conference on Uncertainty Reasoning and Knowledge Engineering (2012).
9.
M.Dasgupta and R.Deb, Transitivity and fuzzy preferences, Social Choice and Welfare13 (1996), 305–318.
10.
S.Diaz, S.Montes and B.De Baets, Transitivity Bounds in Additive Fuzzy Preference Structures, IEEE Transactions on Fuzzy Systems15(2) (2007), 275–286.
11.
S.Diaz, B.De Baets and S.Montes, General results on the decomposition of transitive fuzzy relations, Fuzzy Optimization and Decision Making9(1) (2010), 1–29.
12.
J.C.Fodor and M.R.Roubens, Fuzzy preference modeling and multicriteria decision support (Vol. 14), Springer Science & Business Media (2013).
T.Tanino, Fuzzy preference relations in group decision making, In:Non-conventional preference relations in decision making, Springer, Berlin (1988), 54–71.
16.
E.Trillas and L.Valverde, On some functionally expressible implications for fuzzy set theory, In:3rd International seminar of fuzzy set theory (Johannes Kepler University), Linz (1981), 173–190.
17.
E.Trillas and L.Valverde, On implication and indistin-guishability in the setting of fuzzy logic, In:Management decision support systems using fuzzy set and possibility theory (eds. J.Kacprzyk & R. R.Yager) (1985), pp. 198–212. Verlag TUV Rheinland.
18.
L.Valverde, On the structure of F-indistinguishability operators, Fuzzy Sets and Systems17 (1985), 313–328.
19.
L.A.Zadeh, Fuzzy sets, Information and Control8 (1965), 338–353.
20.
L.A.Zadeh, Similarity relations and fuzzy orderings, Information Sciences3 (1971), 177–200.