The law x ≤ I(y; x) and the three main classes of fuzzy implications

Properties valid on classical theory (Boolean laws) have been extended to fuzzy set theory. Such generalizations of Boolean laws (Boolean-like laws) are not always valid in any standard fuzzy set theory and this fact induced a wide investigation. In this paper we show the conditions that the Boolean-like law x ≤ I(y; x) holds in fuzzy logic. We focus the investigation on three main classes of fuzzy implications, namely: (S,N)-, R- and QL-implications. Further, we show that the operator I satisfies this Boolean-like law if, and only if, Φ-conjugate of I also satisfies it.