On two distributivity equations for fuzzy implications and continuous, Archimedean t-norms and t-conorms (Q691711)

Several classical tautologies from propositional logic were rewritten into the framework of the \([0,1]\) truth-value range, replacing conjunction, disjunction, complement and implication by triangular norms, triangular conorms, fuzzy negations and fuzzy implications, respectively. Obviously, the validity of such a tautology requires to solve relevant functional equations. This approach was applied in numerous papers, including the present one. Here, the considered basic tautologies are: \[ (p\,\, \&\,\, q) \rightarrow r = (p \rightarrow r)\,\, \text{OR}\,\, (q\rightarrow r) \] and its dual form \[ (p\,\, \text{OR}\,\, q)\rightarrow r = (p \rightarrow r)\,\, \&\,\, (q \rightarrow r), \] rewritten into distributivity equations \[ I(T(x,y),z) = S(I(x,z),I(y,z)),\quad x,y,z\, \in\, [0,1] \] and \[ I(S(x,y),z) = T(I(x,z),I(y,z)),\quad x,y,z\in\, [0,1], \] with \(T\) a given continuous Archimedean triangular norm, \(S\) a given continuous Archimedean triangular conorm, and \(I\) an unknown function (in particular, a fuzzy implication). The author characterizes all solutions, considering four basic cases (describing, up to isomorphism, all possible situations) for \(T\) and \(S\). As a by-product, some previous results known from the literature are generalized.