On uninorms with strict underlying t-norm and t-conorm

In this paper we study uninorms with given strict underlying t-norm and t-conorm. Such uninorms are the only candidates for being represented by (additive) generator functions. We prove that representability depends exclusively on the value of the uninorm at a single point in the ldquoremainingrdquo open part of the unit square. The uninorm turns out to be representable if and only if this single value is strictly between the two bounds min and max. If this single value equals to one of the bounds then all values are equal to the same bound at any point in that ldquoremainingrdquo part.