Distributivity of Implication Operations over T-Representable T-Norms Generated from Continuous and Archimedean T-Norms

Abstract

During previous IPMU 2010 conference we have started investigations connected with finding all solutions of the distributive equation of implications $\mathcal{I}(x,\mathcal{T}_1(y,z)) = \mathcal{T}_2(\mathcal{I}(x,y),\mathcal{I}(x,z))$ over t-representable t-norms in interval-valued fuzzy sets theory, i.e., when t-representable t-norms $\mathcal{T}_1$ and $\mathcal{T}_2$ on the lattice $\mathcal{L}^I$ are generated from continuous, Archimedean t-norms T ₁, T ₂ and T ₃, T ₄ on [0,1], respectively. In [2] we have presented solutions when T ₁ = T ₂ = T ₃ = T ₄ is a strict t-norm, in [3] we have discussed the solutions when T ₁ = T ₂ is a nilpotent t-norm and T ₃ = T ₄ is a strict t-norm, while in [4] we have showed the solutions when T ₁ = T ₂ and T ₃ = T ₄ are nilpotent t-norms. In this article we will present the solutions for the last possible case for continuous Archimedean t-norms, i.e. when T ₁ = T ₂ is a strict t-norm and T ₃ = T ₄ is a nilpotent t-norm. As a byproduct result we show all solutions of some functional equation related to this case.