On the distributivity of implication operators over T and S norms

Abstract

In this paper, we explore the distributivity of implication operators [especially Residuated (R)- and Strong (S)-implications] over Takagi (T)- and Sugeno (S)-norms. The motivation behind this work is the on going discussion on the law [(p/spl and/q)/spl rarr/r]/spl equiv/[(p/spl rarr/r)/spl or/(q/spl rarr/r)] in fuzzy logic as given in the title of the paper by Trillas and Alsina. The above law is only one of the four basic distributive laws. The general form of the previous distributive law is J(T(p,q),r)/spl equiv/S(J(p,r),J(q,r)). Similarly, the other three basic distributive laws can be generalized to give equations concerning distribution of fuzzy implications J on T- and S- norms. In this paper, we study the validity of these equations under various conditions on the implication operator J. We also propose some sufficiency conditions on a binary operator under which the general distributive equations are reduced to the basic distributive equations and are satisfied. Also in this work, we have solved one of the open problems posed by M. Baczynski (2002).