## The distributivity condition for uninorms and t-operators.(English)Zbl 1005.03047

Fuzzy Sets Syst. 128, No. 2, 209-225 (2002); corrigendum ibid. 153, No. 2, 297-299 (2005).
Summary: In this work we study the functional equation given by the distributivity condition: $F\bigl(x,G(y,z) \bigr)= G\bigl(F (x,y), F(x,z)\bigr) \text{ for all }x,y,z \in[0,1],$ where the unknown functions $$F,G$$ are uninorms and/or t-operators. We characterize all the solutions in the four possible cases: (i) when $$F,G$$ are t-operators, (ii) when $$F$$ is a t-operator and $$G$$ a uninorm, (iii) when $$F$$ is a uninorm and $$G$$ a t-operator and, finally, (iv) when $$F,G$$ are uninorms.

### MSC:

 3e+72 Theory of fuzzy sets, etc.

Zbl 0996.03038
Full Text:

### References:

 [1] Alsina, C.; Mayor, G.; Tomás, M. S.; Torrens, J., Subdistributive De Morgan triplets, Serdica, 19, 258-266 (1993) · Zbl 0826.04003 [2] Alsina, C.; Trillas, E., On almost distributive Łukasiewicz triplets, Fuzzy Sets and Systems, 50, 2, 175-178 (1992) · Zbl 0782.04006 [3] Calvo, T., On some solutions of the distributivity equation, Fuzzy Sets and Systems, 104, 1, 85-96 (1999) · Zbl 0928.03023 [4] Calvo, T.; De Baets, B., On the generalization of the absortion equation, J. Fuzzy Math., 8, 1, 141-149 (2000) · Zbl 0953.03063 [5] Calvo, T.; De Baets, B., J. Fodor, The functional equations of Frank and Alsina for uninorms and nullnorms, Fuzzy Sets and Systems, 120, 385-394 (2001) · Zbl 0977.03026 [6] T. Calvo, A. Fraile, G. Mayor, Algunes consideracions sobre connectius generalitzats, Actes del VI Congrés Català de Lògica, Barcelona, 1986, pp. 45-46.; T. Calvo, A. Fraile, G. Mayor, Algunes consideracions sobre connectius generalitzats, Actes del VI Congrés Català de Lògica, Barcelona, 1986, pp. 45-46. [7] Carbonell, M.; Mas, M.; Suñer, J.; Torrens, J., On distributivity and modularity in De Morgan triplets, Internat. J. Uncertainty, Fuzziness, Knowledge-based Systems, 4, 4, 351-368 (1996) · Zbl 1232.03038 [8] De Baets, B., Idempotent uninorms, European J. Oper. Res., 118, 3, 631-642 (1999) · Zbl 0933.03071 [9] B. De Baets, Uninorms: the known classes, in: D. Ruan, H.A. Abderrahim, P. D’hondt, E.E. Kerre (Eds.), Proceedings of the Third International FLINS Workshop on Fuzzy Logic and Intelligent Technologies for Nuclear Science and Industry, Antwerp, Belgium, World Scientific, Singapore, 1998.; B. De Baets, Uninorms: the known classes, in: D. Ruan, H.A. Abderrahim, P. D’hondt, E.E. Kerre (Eds.), Proceedings of the Third International FLINS Workshop on Fuzzy Logic and Intelligent Technologies for Nuclear Science and Industry, Antwerp, Belgium, World Scientific, Singapore, 1998. [10] Fodor, J. C.; Yager, R. R.; Rybalov, A., Structure of uninorms, Internat. J. Uncertainty, Fuzziness Knowledge-based Systems, 5, 4, 411-427 (1997) · Zbl 1232.03015 [11] Mas, M.; Mayor, G.; Torrens, J., $$t$$-operators, Internat. J. Uncertainty, Fuzziness Knowledge-based Systems, 7, 1, 31-50 (1999) · Zbl 1087.03515 [12] M. Mas, G. Mayor, J. Torrens, $$t$$; M. Mas, G. Mayor, J. Torrens, $$t$$ · Zbl 0948.68173 [13] Mas, M.; Mayor, G.; Torrens, J., The modularity condition for uninorms and $$t$$-operators, Fuzzy Sets and Systems, 126, 207-218 (2002) · Zbl 0996.03038 [14] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), Elsevier: Elsevier New York · Zbl 0546.60010 [15] Yager, R. R.; Rybalov, A., Uninorm aggregation operators, Fuzzy Sets and Systems, 80, 111-120 (1996) · Zbl 0871.04007 [16] Zimmermann, H. J.; Zysno, P., Latent connectives in human decision making, Fuzzy Sets and Systems, 4, 37-51 (1980) · Zbl 0435.90009 [17] Zimmermann, H. J.; Zysno, P., Decisions and evaluations by hierarchical aggregation of information, Fuzzy Sets and Systems, 10, 243-260 (1983) · Zbl 0519.90049
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