Web geometrical View of Triangular norms and Subnorms

Abstract

The monotonicity, commutativity and neutrality of $1$ are defining properties of triangular norms which are all immediatelly visible on their surface plots. On the other hand, the associativity seems to resist all intuitive geometric interpretations. The less intuitive and more technical aproaches are possible but rare. To our best knowledge, within the framework of triangular norms, these are only the rotation-annihilation aproach by Jenei, the approach of De Baets and Maes based on a contour-view of triangular norms as well as their approach based on the tripple rotation, and, finally, the aproach of Vetterlein based on a sector-view of triangular norms. It is not very well known that the web geometry, which is usually understood to be a branch of differential geometry, provides several geometric concepts which are closely tied to plethora of algebraic properties (such as, among others, associativity) of loops. We realized that, even though triangular norms are not loops, the web-geometric concepts could be modified easily to fit our framework. As a proof of concept, we managed to give the definitive negative answer to the question of convex combinations of nilpotent continuous triangular norms. Now we examine the structure of positively cancellativecontinuous triangular subnorms. As the main result of this study, we provide a structural theory of continuous positively continuous triangular subnorms, i.e., their simple and constructive characterization.