On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums

Uninorms as binary operations on the unit interval have been widely applied in the fuzzy set theory. This paper presents some properties of uninorm-like operations for which the underlying operations are given by ordinal sums. If the underlying operations of a uninorm are given by ordinal sums, then the Cartesian product of the union of two arbitrary intervals (one in [0,e] and the other in [e,1], where e is the neutral element of the uninorm) is a set closed under the uninorm. When transposing such a Cartesian product to the unit square, one obtains a uninorm-like operation. As a result, we have described the uninorm-like operations for which the underlying operations are basic pseudo-t-norms and pseudo-t-conorms and one of them is idempotent.