Preassociative aggregation functions

The classical property of associativity is very often considered in aggregation function theory and fuzzy logic. In this paper we provide axiomatizations of various classes of preassociative functions, where preassociativity is a generalization of associativity recently introduced by the authors. These axiomatizations are based on existing characterizations of some noteworthy classes of associative operations, such as the class of Acz\'elian semigroups and the class of t-norms.Comment: arXiv admin note: text overlap with arXiv:1309.730

Fuzzy Implications: Some Recently Solved Problems

In this chapter we discuss some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. In fact, this chapter also contains the answer for one of the open problems, which is hitherto unpublished. The recently solved problems are so chosen to reflect the importance of the problem or the significance of the solution. Finally, some other problems that still remain unsolved are stated for quick reference

Fuzzy measures assuming their values in the set of fuzzy numbers

AbstractFuzzy-valued fuzzy measures are defined in an axiomatic way. Extending a result of Klement (J. Math. Anal. Appl. 75 (1980), 330–339) it is shown that they can be characterized by a suitable family of ordinary measures and Markov kernels

Characterization of fuzzy measures constructed by means of triangular norms

AbstractFollowing the ideas presented by the author (E. P. Klement, J. Math. Anal. Appl. 85 (1982), 543–565) finite T-fuzzy measures are introduced, T being a measurable triangular norm. We show that a T-fuzzy measure is always a fuzzy measure, as considered earlier (E. P. Klement, J. Math. Anal. Appl. 25 (1980), 330–339). Then we study the relation to the integral with respect to some classical measure. Finally, for some special triangular norms T, we give precise characterizations of the corresponding classes of T-fuzzy measures