Compositions of fuzzy implications

Abstract

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 â̂ - and discuss the dependencies between algebraic properties of the operation â̂ - and the induced sup - â̂ - composition. Under some simple assumptions the sup - â̂ - 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. © Springer-Verlag Berlin Heidelberg 2013.