Previous |  Up |  Next


category theory; aggregation operator; associative aggregation operator; partially ordered groupoid; partially ordered semigroup; partially ordered monoid
In spite of increasing studies and investigations in the field of aggregation operators, there are two fundamental problems remaining unsolved: aggregation of $L$-fuzzy set-theoretic notions and their justification. In order to solve these problems, we will formulate aggregation operators and their special types on partially ordered sets with universal bounds, and introduce their categories. Furthermore, we will show that there exists a strong connection between the category of aggregation operators on partially ordered sets with universal bounds (Agop) and the category of partially ordered groupoids with universal bounds (Pogpu). Moreover, the subcategories of Agop consisting of associative aggregation operators, symmetric and associative aggregation operators and associative aggregation operators with neutral elements are, respectively, isomorphic to the subcategories of Pogpu formed by partially ordered semigroups, commutative partially ordered semigroups and partially ordered monoids in the sense of Birkhoff. As a justification of the present notions and results, some relevant examples for aggregations operators on partially ordered sets are given. Particularly, aggregation process in probabilistic metric spaces is also considered.
[1] Adámek J., Herrlich, H., Strecker G. E.: Abstract and Concrete Categories. Wiley, New York 1990 MR 1051419 | Zbl 1113.18001
[2] Atanassov K. T.: Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20 (1986), 87–96 DOI 10.1016/S0165-0114(86)80034-3 | MR 0852871 | Zbl 0631.03040
[3] Birkhoff G.: Lattice Theory. Third edition. Amer. Math. Soc. Colloquium Publications, Amer. Math. Soc., Providence, RI 1967 MR 0227053 | Zbl 0537.06001
[4] Calvo T., Mayor, G., (eds.) R. Mesiar: Aggregation Operators. New Trends and Applications. (Studies in Fuzziness and Soft Computing, Vol. 97.) Physica-Verlag, Heidelberg 2002 MR 1936383 | Zbl 0983.00020
[5] Deschrijver G., Kerre E. E.: On the relationship between some extensions of fuzzy set theory. Fuzzy Sets and Systems 133 (2003), 227–235 MR 1949024 | Zbl 1013.03065
[6] Deschrijver G., Kerre E. E.: Implicators based on binary aggregation operators in interval-valued fuzzy set theory. Fuzzy Sets and Systems 153 (2005), 229–248 MR 2150282 | Zbl 1090.03024
[7] Deschrijver G., Kerre E. E.: Triangular norms and related operators in L$^{\ast }$-fuzzy set theory. In: Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms (E. P. Klement and R. Mesiar, eds.), Elsevier, Amsterdam 2005, pp. 231–259 MR 2165237 | Zbl 1079.03043
[8] Dubois D., Prade H.: Fuzzy Sets and Systems. Theory and Applications. Academic Press, New York 1980 MR 0589341 | Zbl 0444.94049
[9] Fodor J. C., Yager R. R., Rybalov A.: Structure of uninorms. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 5 (1997), 411–427 DOI 10.1142/S0218488597000312 | MR 1471619 | Zbl 1232.03015
[10] Höhle U., (eds.) S. E. Rodabaugh: Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory. Kluwer Academic Publishers, Boston 1999 MR 1788899 | Zbl 0942.00008
[11] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Vol. 8 of Trends in Logic. Kluwer Academic Publishers, Dordrecht 2000 MR 1790096 | Zbl 1087.20041
[12] Lázaro J., Calvo T.: XAO Operators – The interval universe. In: Proc. 4th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2005), pp. 198–203
[13] Mizumoto M., Tanaka K.: Some properties of fuzzy sets of type 2. Inform. and Control 31 (1976), 312–340 DOI 10.1016/S0019-9958(76)80011-3 | MR 0449947 | Zbl 0331.02042
[14] Sambuc R.: Fonctions $\Phi $-floues. Application à l’aide au diagnostic en pathologie thyroidienne, Ph. D. Thesis, Université de Marseille 1975
[15] 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
[16] Schweizer B., Sklar A.: Probabilistic Metric Spaces. North–Holland, New York 1983 MR 0790314 | Zbl 0546.60010
[17] Zadeh L. A.: Fuzzy sets. Inform. and Control 8 (1965), 338–353 DOI 10.1016/S0019-9958(65)90241-X | MR 0219427 | Zbl 0139.24606
Partner of
EuDML logo