On Linear and Quadratic Constructions of Fuzzy Implication Functions

Abstract

In this paper a new construction method of fuzzy implication functions from a given one, based on ternary polynomial functions is presented. It is proved that the case of linear polynomial functions leads only to trivial solutions and thus the quadratic case is studied in depth. It is shown that the quadratic method allows many different possibilities depending on the usual properties of fuzzy implications functions that we want to preserve. Specifically, there are infinitely many quadratic functions that transform fuzzy implication functions satisfying properties like the neutrality principle, the identity principle, or the law of contraposition with respect to the classical negation, into new fuzzy implication functions satisfying them.