From: ulrich.bodenhofer@scch.at   
      
   > how is the transitivity (max-min transitivity) defined on a fuzzy set X   
   > and with a fuzzy relation R?   
   >   
   I have never heard of a definition of transitivity on a _fuzzy_ set. Would   
   this make sense?   
      
   Anyway, the standard definition, with X being an ordinary non-empty crisp   
   set, is the following:   
      
   **   
   A fuzzy relation R: X x X -> [0,1] is called _min-transitive_ if and only if   
   the following inequality holds for all triples x,y,z of elements from X:   
      
      min(R(x,y),R(y,z)) <= R(x,z)   
   **   
      
   Be aware that min-transitivity is a very strong (most often too strong)   
   requirement - for several deep reasons that I do not want to go into detail   
   about. It is more common to assume that transitivity is considered with   
   respect to some triangular norm that is smaller than the minimum (e.g.   
   product, Lukasiewicz t-norm, etc.).   
      
   Then the definition has to be rephrased accordingly:   
      
   **   
   A fuzzy relation R: X x X -> [0,1] is called _T-transitive_ (with T being a   
   given t-norm) if and only if the following inequality holds for all triples   
   x,y,z of elements from X:   
      
      T(R(x,y),R(y,z)) <= R(x,z)   
   **   
      
   Please find enclosed a few references that might be helpful for you.   
      
   Best regards,   
   Ulrich   
      
      
   --   
      
      
   @book{Belohlavek02,   
     author =  {R. {B\v{e}lohl\'avek}},   
     title =  {Fuzzy Relational Systems. Foundations and   
                     Principles},   
     publisher =  {Kluwer Academic/Plenum Publishers},   
     series =  {IFSR Int. Series on Systems Science and Engineering},   
     address =  {New York},   
     year =  {2002}   
   }   
      
   @incollection{BoixaderJacasRecasens00,   
     author =  {D. Boixader and J. Jacas and J. Recasens},   
     title =  {Fuzzy Equivalence Relations: Advanced Material},   
     booktitle =  {Fundamentals of Fuzzy Sets},   
     editor =  {D. Dubois and H. Prade},   
     publisher =  {Kluwer Academic Publishers},   
     address =  {Boston},   
     volume =  {7},   
     series =  {The Handbooks of Fuzzy Sets},   
     year =  {2000},   
     pages =  {261--290}   
   }   
      
   @book{FodorRoubens94,   
     author =  {J. Fodor and M. Roubens},   
     title =  {Fuzzy Preference Modelling and Multicriteria   
                     Decision Support},   
     year =  {1994},   
     publisher =  {Kluwer Academic Publishers},   
     address =  {Dordrecht}   
   }   
      
   @book{Gottwald93,   
     author =  {S. Gottwald},   
     title =  {Fuzzy Sets and Fuzzy Logic},   
     publisher =  {Vieweg},   
     address =  {Braunschweig},   
     year =  {1993}   
   }   
      
   @book{Hajek98,   
     author =  {P. H\'ajek},   
     title =  {Metamathematics of Fuzzy Logic},   
     publisher =  {Kluwer Academic Publishers},   
     volume =  {4},   
     series =  {Trends in Logic},   
     address =  {Dordrecht},   
     year =  {1998}   
   }   
      
   @article{KlawonnCastro95,   
     author =  {F. Klawonn and J. L. Castro},   
     title =  {Similarity in Fuzzy Reasoning},   
     journal =  {Mathware Soft Comput.},   
     year =  {1995},   
     volume =  {3},   
     number =  {2},   
     pages =  {197--228}   
   }   
      
   @book{KlementMesiarPap00,   
     author =  {E. P. Klement and R. Mesiar and E. Pap},   
     title =  {Triangular Norms},   
     publisher =  {Kluwer Academic Publishers},   
     address =  {Dordrecht},   
     volume =  {8},   
     series =  {Trends in Logic},   
     year =  {2000}   
   }   
      
   @book{KruseGebhardtKlawonn94,   
     author =  {R. Kruse and J. Gebhardt and F. Klawonn},   
     title =  {Foundations of Fuzzy Systems},   
     publisher =  {John Wiley \& Sons},   
     address =  {New York},   
     year =  {1994}   
   }   
      
   @book{NguyenWalker97,   
     author =  {H. T. Nguyen and E. Walker},   
     title =  {A First Course in Fuzzy Logic},   
     publisher =  {CRC Press},   
     address =  {Boca Raton},   
     year =  {1997}   
   }   
      
   @article{Ovchinnikov91,   
     author =  {S. V. Ovchinnikov},   
     title =  {Similarity relations, fuzzy partitions, and fuzzy   
                     orderings},   
     journal =  {Fuzzy Sets and Systems},   
     year =  {1991},   
     volume =  {40},   
     number =  {1},   
     pages =  {107--126}   
   }   
      
   @incollection{Ovchinnikov00,   
     author =  {S. Ovchinnikov},   
     title =  {An Introduction to Fuzzy Relations},   
     booktitle =  {Fundamentals of Fuzzy Sets},   
     editor =  {D. Dubois and H. Prade},   
     publisher =  {Kluwer Academic Publishers},   
     address =  {Boston},   
     volume =  {7},   
     series =  {The Handbooks of Fuzzy Sets},   
     year =  {2000},   
     pages =  {233--259}   
   }   
      
   @incollection{TrillasValverde84,   
     author =  {E. Trillas and L. Valverde},   
     title =  {An Inquiry into Indistinguishability Operators},   
     editor =  {H. J. Skala and S. Termini and E. Trillas},   
     pages =  {231--256},   
     booktitle =  {Aspects of Vagueness},   
     publisher =  {Reidel},   
     address =  {Dordrecht},   
     year =  {1984}   
   }   
      
   @article{Zadeh71,   
     author =  {L. A. Zadeh},   
     title =  {Similarity Relations and Fuzzy Orderings},   
     journal =  {Inform. Sci.},   
     volume =  {3},   
     pages =  {177--200},   
     year =  {1971}   
   }   
      
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