Formula omitted]-implications and distributivity of implications on [formula omitted] over t-representable t-norms: The case of strict and nilpotent t-norms

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In this paper, a new class of intuitionistic fuzzy implications (IFIs) known as (fI,ω)-implications is introduced which is a generalized form of Yager’s f-implications in intuitionistic fuzzy environment (IFE). The basic properties of these implications are discussed in detail. It is shown that (fI,ω)-implications are not only the generalizations of Yager’s f-implications, but also the generalizations of R-, (S,N)- and QL-implications in IFE. The distributivity equations II(T(u,v),w)=S(II(u,w),II(v,w)) and II(u,T1(v,w))=T2(II(u,v),II(u,w)) over t-representable t-norms and t-conorms generated from nilpotent and strict t-norms in IF set theory are discussed. Also, we solve the open problems concerning characterize all of the correct solutions of the distributive equation II(u,T1(v,w))=T2(II(u,v),II(u,w)) when t-norms are strict and nilpotent.