https://www.math.sk/fsta2012/presentations/matusiewicz-drewniak.pdf

Increasing, continuous operations in fuzzy max equations and inequalities

Properties of Atanassov's intuitionistic fuzzy relations and Atanassov's operators

We extend the result of Zhang et al. (J Fuzzy Math 14:53, 2006), who discussed the finite fuzzy relation equations with max–min and max–prod composition. In this article, the $$\text{max-}*$$ composition is used for wide family of operations $$*$$. In particular, families of solutions of two relation equations are compared.

Properties of $$\text{max-}*$$fuzzy relation equations

The goal of this paper is to consider properties of the composition of interval-valued fuzzy relations which were introduced by L.A. Zadeh in 1975. Fuzzy set theory turned out to be a useful tool to describe situations in which the data are imprecise or vague. Interval-valued fuzzy set theory is a generalization of fuzzy set theory which was introduced also by Zadeh in 1965. This paper generalizes some properties of interval matrices considered by Pekala (2007) on these of interval-valued fuzzy relations.

Preservation of properties of interval-valued fuzzy relations

This chapter considers fuzzy implications as a special case of fuzzy relations in [0,1]. We examine compositions of fuzzy implications based on a binary operation ∗ and discuss the dependencies between algebraic properties of the operation ∗ and the induced sup − ∗ composition. Under some simple assumptions the sup − ∗ composition of fuzzy implications gives also a fuzzy implication. This leads to an examination of ordered groupoids and ordered semigroups of fuzzy implications. Contrapositive and invariant fuzzy implications are also considered.

Compositions of Fuzzy Implications

We examine compositions of fuzzy relations based on a binary operation *. We discuss the dependences between algebraic properties of the operation * and the induced sup - * composition. It is examined independently for monotone operations, for operations with idempotent, zero or identity element, for distributive and associative operations. Finally, we present consequences of these results for compositions based on triangular norms, triangular conorms and uninorms.

Krzysztof Kula

Papers

Generalized compositions of fuzzy relations