A characterization of a class of uninorms with continuous underlying operators

In this paper all uninorms locally internal in the region A(e)A(e) (given by the complement in [0,1]2[0,1]2 of [0,e]2∪[e,1]2[0,e]2∪[e,1]2, where e   is the neutral element of the uninorm) having continuous underlying operators are studied and characterized, by distinguishing some cases. When the underlying t-norm and t-conorm are not given by ordinal sums, it is proved that uninorms locally internal in A(e)A(e) are in fact all possible uninorms with these underlying operators (except when both the t-norm and the t-conorm are strict in which case there is also the class of representable uninorms), leading to a finite number of possibilities. When at least one of the continuous underlying operators is given by an ordinal sum, again there are other possible uninorms than those that are locally internal in A(e)A(e), but all uninorms with this property are also characterized. In this case, infinitely many possibilities can appear depending on the set of idempotent elements of the uninorm.