On α-Migrativity of Fuzzy Implications and the Generalised Laws of Importation

In this work, we discuss the law of alpha-migrativity as applied to fuzzy implication functions in a meaningful way. A generalisation of this law leads us to pexider-type generalisations of the law of importation, viz., the generalised law of importation I(C(alpha, x), y) = I(x, J (alpha, y)) (GLI) and the generalised cross-law of importation I (C (alpha, x), y) = J (x, I (alpha, y)) (CLI), where C is a generalised conjunction. We firstly show that the satisfaction of law of importation by the pairs (C, I) and/or (C, J) does not necessarily lead to the satisfaction of (GLI). Hence, we study the conditions under which these three laws may be related.