On the distributivity of uninorms over nullnorms

Authors:
Aifang Xie;Huawen Liu
Affiliations:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China;School of Mathematics, Shandong University, Jinan, Shandong 250100, China
Venue:
Fuzzy Sets and Systems
Year:
2013

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Abstract

This paper investigates the functional equation of distributivity of uninorms over nullnorms. We consider the case where the uninorm is continuous in (0,1)^2 or is representable. It has been proved that the absorbing element of the nullnorm is an idempotent element of the uninorm if the distributive equation holds. And then combining the structures of the uninorm and the nullnorm, we give the characterization of the pair of the functions of the uninorm and the nullnorm. Moreover, when the nullnorm is continuous, we obtain the sufficient and necessary conditions under which the distributive equation holds.