On two distributivity equations for fuzzy implications and continuous, Archimedean t-norms and t-conorms

Recently, we have examined the solutions of the following distributivity functional equation I ( x , S 1 ( y , z ) ) = S 2 ( I ( x , y ) , I ( x , z ) ) , when S 1 , S 2 are continuous, Archimedean t-conorms and I is an unknown function. In particular, between these solutions, we have shown that implication functions are among its solutions. In this paper we continue these investigations for the following distributivity equations I ( T ( x , y ) , z ) = S ( I ( x , z ) , I ( y , z ) ) , I ( S ( x , y ) , z ) = T ( I ( x , z ) , I ( y , z ) ) , when T is a continuous, Archimedean t-norm and S is a continuous, Archimedean t-conorm. The first equation has been investigated by Trillas and Alsina in 2002 [31], while the second equation has been investigated by Balasubramaniam and Rao in 2004 [12], for different classes of fuzzy implications, like R-implications, S-implications and QL-implications. Obtained results are not only theoretical but it can also be useful for the practical problems, since such equations have an important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems.