@ARTICLE{FUZNORM1.MIZ,
  AUTHOR = {Grabowski, Adam},
  TITLE = {Basic Formal Properties of Triangular Norms and Conorms},
  DAY = {27},
  MONTH = {June},
  YEAR = {2017},
  ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
  SECTION1 = {Preliminaries},
  SECTION2 = {Basic Example of a Triangular Norm and Conorm: min and max},
  SECTION3 = {Further Examples of Triangular Norms},
  SECTION4 = {Basic Triangular Conorms},
  SECTION5 = {Translating between Triangular Norms and Conorms},
  SECTION6 = {The Ordering of Triangular Norms (and Conorms)},
  SECTION7 = {Triangular Conorms Generated from t-Norms},
  SUMMARY = {In the article we present in the Mizar system
  \cite{Mizar-State-2015} the catalogue of triangular norms and conorms,
  used especially in the theory of fuzzy sets \cite{Zadeh:1965}. The name
  \emph{triangular} emphasizes the fact that in the framework of
  probabilistic metric spaces they generalize triangle inequality.\par
  We introduced the following t-norms:
  \begin{itemize}
  \item minimum t-norm \verb!minnorm!,
  \item product t-norm \verb!prodnorm!,
  \item \L ukasiewicz t-norm \verb!Lukasiewicz_norm!,
  \item drastic t-norm \verb!drastic_norm!,
  \item nilpotent minimum \verb!nilmin_norm!,
  \item Hamacher product \verb!Hamacher_norm!,
  \end{itemize}
  and corresponding t-conorms:
  \begin{itemize}
  \item maximum t-conorm \verb!maxnorm!,
  \item probabilistic sum \verb!probsum_conorm!,
  \item bounded sum \verb!BoundedSum_conorm!,
  \item drastic t-conorm \verb!drastic_conorm!,
  \item nilpotent maximum \verb!nilmax_conorm!,
  \item Hamacher t-conorm \verb!Hamacher_conorm!.
  \end{itemize}
  Their basic properties and duality are shown; we also
  proved the predicate of the ordering of norms \cite{Klement:2000},
  \cite{Hajek:1998}. This work is a continuation of the development
  of fuzzy sets in Mizar \cite{GrabowskiFuzzy:2013};
  it could be used to give a variety of
  more general operations on fuzzy sets.},}

@BOOK{Klement:2000,
AUTHOR = {Klement, Erich Peter and Mesiar, Radko and Pap, Endre},
TITLE = {Triangular Norms},
PUBLISHER = {Dordrecht: Kluwer},
YEAR = {2000}}

@BOOK{Hajek:1998,
AUTHOR = {H\'ajek, Petr},
TITLE = {Metamathematics of Fuzzy Logic},
PUBLISHER = {Dordrecht: Kluwer},
YEAR = {1998}}
