Kybernetika 43 no. 2, 183-196, 2007

On the structure of continuous uninorms

Paweł Drygaś


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].