The Cai δ-equality of fuzzy sets corresponds to the Lukasiewicz t-norm. In this paper we study the notion of (∗, δ)-equality, a concept which generalizes the δ-equality to the case of the fuzzy set theory based on an arbitrary continuous t-norm ∗. We investigate the robustness of some fuzzy implication operators in terms of (∗, δ)-equality.

In this paper we propose computationally simple, pointwise formulas for extended t-norms and t-conorms on fuzzy truth values. The complex convolutions of the extended operations are shown to be equivalent to simple pointwise expressions for several special cases. Linear fuzzy truth values are defined and it is shown that the extended Łukasiewicz operations preserve linearity. Since linear fuzzy...

In this paper we consider a special class of t-norms TT1,T2,t on the lattice LI , where LI is the underlying lattice of both intuitionistic fuzzy set theory (Atanassov, 1983) and interval-valued fuzzy set theory (Sambuc, 1975). We investigate under which conditions these tnorms are meet-morphisms. Using these results, we obtain a characterization for t-norms on LI which are both joinand meet-mo...

Algebraic structures such as Bottleneck Algebras (R,Max,Min), Fuzzy Algebras ((0, 1),Max,Min) or more generally ((0, 1),Max, T ) where T is a t-norm have been extensively used as relevant tools for modeling and solving problems related to Fuzzy Sets, Fuzzy relations and systems. Many of these algebraic structures may be viewed as special instances of canonically ordered Semirings, (i.e. semirin...

The main subject of this paper is the embedding of fuzzy set theory—and related concepts—in a coherent conditional probability scenario. This allows to deal with perception-based information—in the sense of Zadeh—and with a rigorous treatment of the concept of likelihood, dealing also with its role in statistical inference. A coherent conditional probability is looked on as a general non-additi...

Intuitionistic fuzzy sets in the sense of Atanassov and interval-valued fuzzy sets can be seen as L-fuzzy sets w.r.t. a special lattice L . Deschrijver [2] introduced additive and multiplicative generators on L based on a special kind of addition introduced in [3]. Actually, many other additions can be introduced. In this paper we investigate additive generators on L as far as possible independ...

We define a 2-category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions. We enrich this category with additional monoidal and involutive structure coming from t-norms and negations on the unit interval. We develop the basic properties of this category and consider its relation to other familiar categories. A discussion is made of extending these re...

The notion of filter is established in fuzzy class theory. Graded properties of filters, prime filters and related concepts are investigated. In particular we study properties of filters relativized with respect to the basic three t-norms: Gödel, product, and Lukasiewicz. It is shown that our approach gives rise to a broad class of models including probability and necessity measures on fuzzy se...