# Generators of Fuzzy Logical Operations

Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 878)

## Abstract

We deal with fuzzy logical operations with values in the real unit interval. Many of them can be considered equivalent up to an isomorphism (i.e., increasing bijection) of the set of values. This is the case of all involutive fuzzy negations; an elegant proof was given by Nguyen and Walker (A first course in fuzzy logic, 2nd edn. Chapman & Hall/CRC, Boca Raton, 2000 ) . The situation is more tricky for binary operations, triangular norms, triangular conorms, and fuzzy implications. For the most common classes of these operations, the existence of their (additive or multiplicative) generators is known; however, their computation can be often unfeasible. We proved that a rather general subclass allows computing the generators from partial derivatives. Here we summarize preceding results in this direction (mostly with simplified proofs) and add several new ones.

## Keywords

Additive generator Archimedean triangular conorm Archimedean triangular norm Fuzzy conjunction Fuzzy disjunction Fuzzy logic Fuzzy negation Fuzzy R-implication Fuzzy S-implication Generator Multiplicative generator

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