On the distributivity of fuzzy implications over continuous Archimedean t-conorms and continuous t-conorms given as ordinal sums

In this paper, we investigate the distributive functional equation I ( x , S 1 ( y , z ) ) = S 2 ( I ( x , y ) , I ( x , z ) ) , where I : [ 0,1 ] 2 → [ 0,1 ] is an unknown function, S 2 a continuous Archimedean t-conorm and S 1 a continuous t-conorm given as an ordinal sum. First, based on the special case with one summand in the ordinal sum of S 1 , all the sufficient and necessary conditions of solutions to the distributive equation above are given and the characterization of its continuous solutions is derived. It is shown that the distributive equation does not have continuous fuzzy implication solutions. Subsequently, we characterize its non-continuous fuzzy implication solutions. Finally, it is pointed out that the case with finite summands in the ordinal sum of S 1 is equivalent to the one with one summand.