## A general class of triangular norm-based aggregation operators: Quasi-linear $$T-S$$ operators.(English)Zbl 1006.68160

Summary: This paper generalizes the well-known exponential and linear convex $$T-S$$ aggregation operators into a wider class of compensatory aggregation operators, built as the composition of an arbitrary quasi-linear mean with a t-norm and a t-conorm, which are called quasi-linear $$T -S$$ operators. These new operators are compared with other existing ones, and their main properties, such as the existence of neutral or annihilator elements, are studied. In particular, the self-duality property is investigated, and a characterization of an important family of self-dual quasi-linear $$T -S$$ operators is provided.

### MSC:

 68U35 Computing methodologies for information systems (hypertext navigation, interfaces, decision support, etc.) 68T37 Reasoning under uncertainty in the context of artificial intelligence

### Keywords:

aggregation operators
Full Text:

### References:

 [1] Aczél, J., Lectures on functional equations and their applications, (1966), Academic Press New York · Zbl 0139.09301 [2] T. Calvo, Contribución al estudio de las ternas de De Morgan generalizadas, Ph.D. Thesis, 1989 (in Spanish) [3] Calvo, T.; Mesiar, R., Weighted means based on triangular conorms, International journal of uncertainty, fuzziness and knowledge-based systems, 9, 2, 183-196, (2001) · Zbl 1113.03335 [4] Calvo, T.; Kolesárová, A.; Komornı́ková, M.; Mesiar, R., A review of aggregation operators, Handbook of AGOP’2001, (2001), University of Alcalá Press [5] Dubois, D.; Prade, H., A review of fuzzy set aggregation connectives, Information sciences, 36, 85-121, (1985) · Zbl 0582.03040 [6] Dubois, D.; Prade, H., Criteria aggregation and ranking of alternatives in the framework of fuzzy set theory, (), 209-240 [7] Fodor, J.; Roubens, M., On meaningfulness of means, Journal of computational and applied mathematics, 64, 103-115, (1995) · Zbl 0853.39010 [8] Fodor, J.; Calvo, T., Aggregation functions defined by t-norms and t-conorms, (), 36-48 · Zbl 0904.39016 [9] Frank, M., On the simultaneous associativity of F(x,y) and x+y−F(x,y), Aequationes mathematicae, 19, 194-226, (1979) · Zbl 0444.39003 [10] Gehrke, M.; Walker, C.; Walker, E., Averaging operators on the unit interval, International journal of intelligent systems, 14, 883-898, (1999) · Zbl 0935.03061 [11] Klement, E.; Mesiar, R.; Pap, E., Triangular norms, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0972.03002 [12] Kolesárová, A., Comparison of quasi-arithmetic means, (), 237-240 [13] Luhandjula, M.K., Compensatory operators in fuzzy linear programming with multiple objectives, Fuzzy sets and systems, 8, 245-252, (1982) · Zbl 0492.90076 [14] Mayor, G.; Trillas, E., On the representation of some aggregation functions, (), 110-114 [15] Mesiar, R.; Komornı́ková, M., Triangular norm-based aggregation of evidence under fuzziness, (), 11-35 [16] Mesiar, R., Aggregation operators: some classes and construction methods, (), 707-711 [17] Mizumoto, M., Pictorial representations of fuzzy connectives, part II: cases of compensatory operators and self-dual operators, Fuzzy sets and systems, 32, 45-79, (1989) · Zbl 0709.03524 [18] Moser, B.; Tsiporkova, E.; Klement, E., Convex combinations in terms of triangular norms. A characterization of idempotent bisymmetrical and self-dual compensatory operators, Fuzzy sets and systems, 104, 97-108, (1999) · Zbl 0928.03063 [19] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), North-Holland Amsterdam · Zbl 0546.60010 [20] E. Trillas, Sobre funciones de negación en la teorı́a de los subconjuntos difusos. Stochastica, III-1 (1979) 47-59 (in Spanish); S. Barro et al. (Eds.), Advances of Fuzzy Logic (English version), Universidad de Santiago de Compostela, 1998, pp. 31-43 (reprint) [21] Zimmermann, H.J.; Zysno, P., Latent connectives in human decision-making, Fuzzy sets and systems, 4, 37-51, (1980) · Zbl 0435.90009
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