Abstract:

This paper aims at exploring the following distributivity equation on a finite chain: F(G1(x,y),z)=G2(F(x,z),F(y,z)). We characterize all the solutions of the distributivity equation in the following cases: (a) F is a smooth t-norm and G1=G2 are smooth t-conorms(F is a smooth t-conorm and G1=G2 are smooth tnorms); (b) F is an S-implication (or an Rimplication), G1 is a smooth t-norm and G2 is a smooth t-conorm; (c) F is an S-implication (or an R-implication), G1 is a smooth t-conorm and G2 is a smooth t-norm; (d) F is an S-implication (or an R-implication), G1 and G2 are smooth t-norms; (e) F is an S-implication (or an R-implication), G1 and G2 are smooth t-conorms.

Key words: distributivity equation, tnorm, tconorm, Simplication, Rimplication, finite chain