On the characterizations of fuzzy implications satisfying I(x,I(y,z))=I(I(x,y),I(x,z))

Boolean-like laws is inspired by many works in Fuzzy Logic. In this paper, we investigate the fuzzy generalization of: x→(y→z)≡(x→y)→(x→z) law. More precisely, we will show under which conditions such Boolean-like law holds for the following classes of fuzzy implications: (S,N)-, R-, QL-, D-, (T,N)- and H. To achieve that we introduce a new fuzzy property called Distributivity over De Morgan. Finally, we will show that a fuzzy implication I satisfies such Boolean-like law if, and only if, it is also satisfied by its Φ-conjugate.