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)=(/spl alpha//sup p/+ b/sup -p/-1)/sup -1/p/ if (a/sup -p/+b/sup -p/-1)/spl ges/0,0 otherwise. This paper considers the effects of removing the requirement that (a/sup -p/+b/sup -p/-1)/spl ges/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.