Fuzzy implications: alpha migrativity and 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 functional equations connected with the law of importation, viz., the generalised law of importation I(C(x, alpha), y) = I(x, J(alpha, y)) (GLI) and the generalised cross-law of importation I(C(x, alpha), y) = J(x, I(alpha, 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. (c) 2020 The Author(s). Published by Elsevier Inc.