On distributivity equations of implications over overlap functions and contrapositive symmetry equations of implications

In order to avoid combinatorial rule explosion in fuzzy reasoning, distributive laws related to fuzzy implications and aggregation functions have been widely studied in recent years. In 2017, Qiao and Hu investigated the distributivity equation I(x, O-1(y, z)) = O-2(I(x, y), I(x, z)), when O-1 and O-2 are additively generated overlap functions and I is an unknown function. In this paper, this kind of distributivity equation continues to be studied at the situation that both O-1 and O-2 are multiplicatively generated overlap functions and I is a continuous function. Unfortunately, there is no continuous solution for this distributivity equation that is fuzzy implication, but a characterization for the case that I is continuous except for the point (0, 0) is given. More importantly, some solutions to the system of functional equations consisting of I(x, O-1 (y, z)) = O-2(I(x, y), I(x, z)) and I(x, y) = I(N(y), N(x)) are characterized. Finally, considered any other overlap functions, sufficient and necessary conditions, under which the previous distributivity equation holds when O-1 is an idempotent overlap function, O-2 is an Archimedean overlap function and I is an unknown function, are given. Meanwhile, some solutions to the system of functional equations are also presented.