Improper strong implication

Schweizer and Sklar (1961, 1963) define a family of T-norms from [0,1]X[0,1] to [0,1] using a parameter p as follows: T(a,b)=(α^p+ b^-p-1)^-1p/ if (a^-p+b^-p-1)≥0,0 otherwise. This paper considers the effects of removing the requirement that (a^-p+b^-p-1)≥0. This produces a family of complex improper T-norms, which can be used to define a corresponding family of real improper T-norms ranging from -1 to min(a,b). Improper strong implication functions created using the real improper T-norms support a variant of mode defuzzification, called best kernel defuzzification, with potentially useful properties for fuzzy expert systems.