Preservation of the Ordering Property Under the Quadratic Polynomial Construction of Fuzzy Implication Functions

Abstract

In recent academic studies, new ways of constructing fuzzy implication functions have been developed. One of the proposed constructions is a ternary polynomial one of the form IF(x, y): = F(x, y, I(x, y) ), where F is a polynomial, and I is a fuzzy implication. In the particular case of quadratic polynomials, under additional conditions, construction preserves properties like the left neutrality principle, the law of contraposition, or the identity principle of the initial fuzzy implication I. In this paper, we looked at the preservation of the ordering property under such quadratic polynomial construction. Additionally, we took a look at the case of arbitrary degree polynomials and their preservation of boundary conditions of a fuzzy implication.