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The well-known iterative boolean-like law a →(a → b) = a → b can be generalized to the functional equation I(x, I(x, y)) = I(x, y), where I is a fuzzy implication. In this paper, we discuss an approximation of the equation, I(x, I(x, y)) ≈ I(x, y), i.e., the law is approximately valid. Furthermore, we study the property of approximation preserving with respect to compositions of fuzzy implications. Finally, we give a necessary condition and a sufficient condition for the approximate equation of (S, N)-implications. [ABSTRACT FROM AUTHOR]

The iterative functional equation $ \alpha\rightarrow(\alpha\rightarrow \beta) = \alpha\rightarrow \beta $ and the law of importation $ (\alpha\wedge \beta)\rightarrow \gamma = \alpha\rightarrow (\beta\rightarrow \gamma) $ are two tautologies in classical logic. In fuzzy logics, they are two important properties, and are respectively formulated as $ I(\alpha, \beta) = I(\alpha, I(\alpha, \beta)) $ and $ I(T(\alpha, \beta), \gamma) = I(\alpha, I(\beta, \gamma)) $ where $ I $ is a fuzzy implication and $ T $ is a $ t $-norm. Over the past several years, solutions to these two functional equations involving different classes of fuzzy implications have been studied. However, there are no results about stability study of fuzzy functional equations involving fuzzy implication. This paper discusses fuzzy implications that do not strictly satisfying these equations, but approximately satisfy these equations. Then we establish the Hyers-Ulam stability of the iterative functional equation involving the $ (S, N) $-implication, where the $ (S, N) $-implication is a common class of fuzzy implications generated by a continuous $ t $-conorm $ S $ and a continuous fuzzy negation $ N $. Furthermore, given a fixed $ t $-norm (the minimum $ t $-norm or the product $ t $-norm) the Hyers-Ulam stability of the law of importation involving the $ (S, N) $-implication is studied.