On the structure of continuous uninorms

summary:Uninorms were introduced by Yager and Rybalov [13] as a generalization of triangular norms and conorms. We ask about properties of increasing, associative, continuous binary operation $U$ in the unit interval with the neutral element $e\in [0,1]$. If operation $U$ is continuous, then $e=0$ or $e=1$. So, we consider operations which are continuous in the open unit square. As a result every associative, increasing binary operation with the neutral element $e\in (0,1)$, which is continuous in the open unit square may be given in $[0,1)^2$ or $(0,1]^2$ as an ordinal sum of a semigroup and a group. This group is isomorphic to the positive real numbers with multiplication. As a corollary we obtain the results of Hu, Li [7]

Some Examples of Weak Uninorms

It is proved that, except for the uninorms and the nullnorms, there are no continuous weak uninorms who have no more than one nontrivial idempotent element. And some examples of discontinuous weak uninorms are shown. All of these examples are not n-uninorms, thus not uninorms or nullnorms