Associativity of triangular norms characterized by the geometry of their level sets (Q695244)

A visual interpretation of the associativity property was a challenge for several authors, for example, Jenei, Maes and De Baets, Vetterlein, etc. Based on the ideas of web geometry and its concept of Reidemeister closure condition, the authors succeeded to give such an interpretation for triangular norms, i.e., commutative, associative monotone binary functions on \([0,1]\) with neutral element \(e = 1\). The visual characterization of the associativity of a triangular norm \(T\) is based on the geometry of the corresponding level sets \(T^{-1}(a)\), \(a\) from \([0,1]\). The included figures appropriately illustrate the presented concepts and results. The techniques and ideas of this paper can be applied also in other situations, for example, for a visual interpretation of the associativity of triangular conorms or uninorms.