On the distributivity of fuzzy implications over nilpotent or strict triangular conorms

  • Authors:

  • Michał Baczynki;Balasubramaniam Jayaram

  • Affiliations:

  • Institute of Mathematics, University of Silesia, Katowice, Poland;Department of Mathematics and Computer Sciences, Sri Sathya Sai University, Anantapur, Andhra Pradesh, India

  • Venue:
  • IEEE Transactions on Fuzzy Systems
  • Year:
  • 2009

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Abstract

Recently, many works have appeared in this very journal dealing with the distributivity of fuzzy implications over t-norms and t-conorms. These equations have a very important role to play in efficient inferencing in approximate reasoning, especially fuzzy control systems. Of all the four equations considered, the equation I(x, S1 (y, z)) = S2 (I(x, y), I(x, z)), when S1, S2 are both t-conorms and I is an R-implication obtained from a strict t-norm, was not solved. In this paper, we characterize functions I that satisfy the previous functional equation when S1, S2 are either both strict or nilpotent t-conorms. Using the obtained characterizations, we show that the previous equation does not hold when S1, S2 are either both strict or nilpotent t-conorms, and I is a continuous fuzzy implication. Moreover, the previous equation does not hold when I is an R-implication obtained from a strict t-norm, and S1, S2 are both strict t-conorms, while it holds for an R-implication I obtained from a strict t-norm T if and only if the t-conorms S1 = S2 are Φ-conjugate to the Łukasiewicz t-conorm for some increasing bijection ϕ of the unit interval, which is also a multiplicative generator of T.