On the Generalized Cross-Law of Importation in Fuzzy Logic

Recently, Baczyński et al. introduced two pexider-type generalisations of the law of importation in fuzzy logic, i.e., I(C(x, α), y)=I(x, J(α, y) (GLI) and I(C(x, α), y)=J(x, I(α, y) (CLI), where C is a fuzzy conjunction and I, J are fuzzy implications. However, (CLI) has not been adequately investigated so far. In this paper, we firstly show that (CLI) can be derived from the α-migrativity of an R-implication obtained from an α-migrative t-norm. Secondly, the relationships between the satisfaction of the law of importation (LI) by the pairs (C, I) or (C, J) and the satisfaction of (CLI) by the triple (C, I, J) are studied. Moreover, some necessary conditions of (CLI) are given. Finally, we study (CLI) under three different perspectives.