A single-point characterization of representable uninorms

We study uninorms of which both the underlying t-norm and underlying t-conorm are strict. Such uninorms are the only candidates for being representable by an additive generator. We prove that the representability of such a uninorm depends solely on its value at a single arbitrary point in the ‘remaining’ open part of the unit square. More explicitly, such a uninorm turns out to be representable if and only if this single value is located strictly between the minimum and the maximum of the corresponding arguments. If this single value coincides with one of these bounds, then the value of the uninorm at any point in the ‘remaining’ open part is determined by the same bound.