This paper deals with interval-valued fuzzy sets and I-fuzzy sets, Atanassov's intuitionistic fuzzy sets. They are treated in a systematic way as two, formally equivalent, natural extensions of methods of representing incomplete knowledge about sets. We define and investigate triangular norm-based areas of uncertainty of interval-valued fuzzy sets and I-fuzzy sets, and study some properties of ...

This paper presents a variety of cardinality concepts offered by cardinality theory of triangular norm-based fuzzy sets, including the scalar and the "fuzzy" streams.

Recently, in [4], we have discussed the following distributive equation of implications I(x, T1(y, z)) = T2(I(x, y), I(x, z)) over t-representable t-norms, generated from strict t-norms, in interval-valued fuzzy sets theory. In this work we continue these investigations, but with the assumption that T1 is generated from nilpotent t-norms, while T2 is generated from strict t-norms. As a byproduc...

A family of fuzzification schemes is proposed that can be used to transform cardinality-based similarity measures for ordinary sets into similarity measures for fuzzy sets in a finite universe. These schemes are based on the Frank t-norm family and preserve transitivity in the fuzzification process.

This paper is a contribution to the theory of generalized aggregation operators (shortly gagops) introduced by Takači in [11]. The term generalized refers to the inputs of an aggregation operator (shortly agop), they are a special type of fuzzy sets. In the sequel we study a method of construction of a gagop by means of an arbitrary continuous t-norm, called T -extension. Another construction m...