(fI, ω)-implications and distributivity of implications on L over t-representable t-norms: The case of strict and nilpotent t-norms

In this paper, a new class of intuitionistic fuzzy implications (IFIs) known as (f(I), omega)-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 (f(I), omega)-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 I-I (T (u, v), w) = S (I-I (u, w), I-I (v, w)) and I-I (u, (v, w)) = T-2 (I-I (u, v), I-I (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 I-I(u, T-1 (v, w)) = T-2 (I-I(u, v), I-I (u, w)) when t-norms are strict and nilpotent. (C) 2019 Elsevier Inc. All rights reserved.