Associativity of triangular norms characterized by the geometry of their level sets

0381684 - ÚI 2013 RIV NL eng J - Journal Article
Petrík, Milan - Sarkoci, P.
Associativity of triangular norms characterized by the geometry of their level sets.
Fuzzy Sets and Systems. Roč. 202, 1 September (2012), s. 100-109. ISSN 0165-0114. E-ISSN 1872-6801
R&D Projects: GA ČR GAP202/10/1826
Institutional support: RVO:67985807
Keywords : associativity * contour * level set * Reidemeister closure condition * triangular norm * web geometry
Subject RIV: BA - General Mathematics
Impact factor: 1.749, year: 2012

Associativity of triangular norms is an algebraic property which, unlike for example their commutativity, is usually understood as hardly visually interpretable. This problem has been studied intensively in the last decade and, as a result, geometric symmetries of triangular norms with involutive level sets have been revealed. The presented paper intends to introduce a different approach which gives more general results. The inspiration is taken from web geometry, a branch of differential geometry, and its concept of Reidemeister closure condition which is known to provide a geometric characterization of associativity of loops. The paper shows that this concept can be adopted successfully for triangular norms so that it characterizes their associativity in a similar way. Moreover, the offered adaptation preserves the beneficial transparency and simplicity of the Reidemeister closure condition. This way, a visual characterization of the associativity, based on the geometry of the level sets, is provided for general, continuous, and continuous Archimedean triangular norms.
Permanent Link: http://hdl.handle.net/11104/0212099