The functional equations of Frank and Alsina for uninorms and nullnorms
The aim of this work is to study the functional equations of Frank and Alsina for two classes of commutative, associative and increasing binary operators. The first one is the class of uninorms introduced by Yager and Rybalov. The second one is the class of nullnorms arising from our study of the Frank equation for uninorms. Both classes contain t-norms and t-conorms as special cases. Moreover, the structure of the other uninorms and nullnorms is closely related to t-norms and t-conorms. These observations are the motivation for studying some generalizations of the Frank and Alsina equations. However, it is shown that all considerations lead back to the already known t-norm and t-conorm, solutions. Important consequences in fuzzy preference modelling are pointed out. (C) 2001 Elsevier Science B.V. All rights reserved.
functional equation uninorm nullnorm frank equation t-norm t-conorm alsina equation aggregation operators fuzzy connectives