A note on uninorms with continuous underlying operators

Our investigations are motivated by the fact uninorms as mixed aggregation operations on the unit interval have been widely used in fuzzy set theory. This paper is mainly to characterize the inner structure of uninorms with continuous underlying operators given by ordinal sums, and then presents some of their properties. To be specific, for a uninorm, we describe the relations of its two arbitrary summands and characterize the structure on Cartesian products of its underlying intervals corresponding to those two summands mentioned previously (one in [0,e] and the other in [e,1], where e is the neutral element of the uninorm). These results bring us a step closer to get the full structure of a uninorm with continuous underlying operators.