@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 },
}