Fuzzy implications: alpha migrativity and generalised laws of importation.

In this work, we discuss the law of α -migrativity as applied to fuzzy implication functions in a meaningful way. A generalisation of this law leads us to Pexider-type functional equations connected with the law of importation, viz., the generalised law of importation I (C (x , α) , y) = I (x , J (α , y)) (GLI) and the generalised cross-law of importation I (C (x , α) , y) = J (x , I (α , y)) (CLI), where C is a generalised conjunction. In this article we investigate only (GLI). We begin by showing 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 are related. [ABSTRACT FROM AUTHOR]