# Article

Full entry | PDF   (0.7 MB)
Keywords:
fuzzy relation; binary operation; relation composition; \$\sup \nolimits \$-\$\ast \$ composition; relation powers; relation closure; relation interior
Summary:
Properties of \$\sup \nolimits \$-\$\ast \$ compositions of fuzzy relations were first examined in Goguen [8] and next discussed by many authors. Power sequence of fuzzy relations was mainly considered in the case of matrices of fuzzy relation on a finite set. We consider \$\sup \nolimits \$-\$\ast \$ powers of fuzzy relations under diverse assumptions about \$\ast \$ operation. At first, we remind fundamental properties of \$\sup \nolimits \$-\$\ast \$ composition. Then, we introduce some manipulations on relation powers. Next, the closure and interior of fuzzy relations are examined. Finally, particular properties of fuzzy relations on a finite set are presented.
References:
[1] Birkhoff G.: Lattice Theory. (Colloq. Publ. 25.) American Mathematical Society, Providence, RI 1967 MR 0227053 | Zbl 0537.06001
[2] Cechlárová K.: Powers of matrices over distributive lattices – a review. Fuzzy Sets and Systems 138 (2003), 3, 627–641 MR 1998683 | Zbl 1075.05537
[3] Drewniak J.: Classes of fuzzy relations. In: Application of Logical an Algebraic Aspects of Fuzzy Relations (E. P. Klement and L. I. Valverde eds.), Johannes Kepler Universität Linz, Linz 1990, pp. 36–38
[4] Drewniak J., Kula K.: Generalized compositions of fuzzy relations. Internat. J. Uncertainty, Fuzziness Knowledge-Based Systems 10 (2002), 149–163 MR 1962675 | Zbl 1053.03511
[5] Fan Z. T.: A note on power sequence of a fuzzy matrix. Fuzzy Sets and Systems 102 (1999), 281–286 MR 1674967
[6] Fan Z. T.: On the convergence of a fuzzy matrix in the sense of triangular norms. Fuzzy Sets and Systems 109 (2000), 409–417 MR 1746994 | Zbl 0980.15013
[7] Fodor J., Roubens M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht 1994 Zbl 0827.90002
[8] Goguen J. A.: L-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145–174 MR 0224391 | Zbl 0145.24404
[9] Kaufmann A.: Introduction to the Theory of Fuzzy Subsets. Academic Press, New York 1975 MR 0485402 | Zbl 0332.02063
[10] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000 MR 1790096 | Zbl 1087.20041
[11] Klir G. J., Yuan B.: Fuzzy Sets and Fuzzy Logic. Theory and Applications. Prentice Hall, New Jersey 1995 MR 1329731 | Zbl 0915.03001
[12] Li J. C., Zhang W. X.: On convergence of the min-max compositions of fuzzy matrices. Southeast Asian Bull. Math. 24 (2000), 3, 389–393 MR 1811398 | Zbl 0982.15018
[13] Li J. X.: An upper bound of indices of finite fuzzy relations. Fuzzy Sets and Systems 49 (1992), 317–321 MR 1185382
[14] Nguyen H. T., Walker E. A.: A First Course in Fuzzy Logic. Chapmann & Hall, London 2000 MR 1700266 | Zbl 1083.03031
[15] Portilla M. I., Burillo, P., Eraso M. L.: Properties of the fuzzy composition based on aggregation operators. Fuzzy Sets and Systems 110 (2000), 2, 217–226 MR 1747743 | Zbl 0941.03060
[16] Thomason M. G.: Convergence of powers of a fuzzy matrix. J. Math. Anal. Appl. 57 (1977), 476–480 MR 0427342 | Zbl 0345.15007
[17] Szász G.: Introduction to Lattice Theory. Akad. Kiadó, Budapest 1963 MR 0110652 | Zbl 0126.03703
[18] Tan Y. J.: On the transitive matrices over distributive lattices. Linear Algebra Appl. 400 (2005), 169–191 MR 2131923 | Zbl 1073.15015
[19] Zadeh L. A.: Fuzzy sets. Inform. and Control 8 (1965), 338–353 MR 0219427 | Zbl 0139.24606
[20] Zadeh L. A.: Similarity relations and fuzzy orderings. Inform. Sci. 3 (1971), 177–200 MR 0297650 | Zbl 0218.02058

Partner of