@InProceedings{Petrik:DD2009, IS = { zkontrolovano 29 Jan 2010 }, UPDATE = { 2009-09-29 }, author = { Petr{\'\i}k, Milan }, title = { Properties of Fuzzy Logical Operations }, year = { 2009 }, pages = { 89--96 }, booktitle = { Doktorandsk{\'e} dny 2009 {\'U}stavu informatiky AV {\v C}R, v. v. i. }, editor = { Ku{\v z}elov{\'a}, D. }, publisher = { Matfyzpress }, address = { Prague, Czech Republic }, isbn = { 978-80-7378-087-6 }, book_pages = { 133 }, month = { September }, day = { 21--23 }, venue = { Jizerka, Czech Republic }, organization ={ Institute of Computer Science, Academy of Sciences of the Czech Republic }, annote = { We deal with geometrical and differential properties of triangular norms (t-norms for short), i.e. binary operations which implement logical conjunctions in fuzzy logic. The first part discusses the problem of a visual characterization of the associativity of t-norms. The results given by web geometry are adopted, mainly the concept of the Reidemeister closure condition, in order to characterize the shape of level sets of t-norms. This way, a visual characterization of the associativity is provided for general, continuous, and continuous Archimedean t-norms. The second part deals with differential properties of continuous Archimedean t-norms. It is shown that partial derivatives of such a t-norm on a particular subset of its domain correspond directly to the generator (or to the derivative of the generator) of the t-norm. As the result, several methods which reconstruct multiplicative and additive generators of continuous Archimedean t-norms are introduced. The presented results contribute to a partial solution of an open problem whether a non-trivial convex combination of two t-norms can be a triangular norm again. }, keywords = { triangular norm, web geometry, generator, reconstruction, convex combination }, prestige = { local }, authorship = { 100 }, project = { GACR 401/09/H007 }, }