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Some properties concerning the quasi-inverse of a t-norm

Boixader Ibáñez, Dionís

Abstract

Some properties of the quasi-inverse operators are presented. They are basic tools in order to reduce complex expressions involving several of such operators. An effective calculation for the quasi-inverse of a continuous t-norm is also provided.

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                             !  "#  "$                                                                                                                                    !  "#  "$%&             '%#  #() *+,&                       - -       %              *.  /,  *.  /,  *.  /,              /  &"    .  &".     *.  /,           0    0     & "     *.  /,    &          *.  /,  1 2                     &"   0     &  0     &  " 0            &      *.  /,  3         44              .  /  & 5       0                      0     &          &       & 6              0     &        *1,    *2,        4      44 ! 3        &     44    & )                  0       7         /&    & " #       0     &"  #          /     8&    & " $%   & " #   9   /  .    %!4 &  0     &"  /   9      /     +&    &"     0     &"       /     :&    &"    &"       "/    "/ .     0    &"  /  /    "/              &       .  /&            &  .     .  /,            &         &"      &"        &&    /  ;   4      %&                  *.  /,  *.  9  ,     /& " .     &"       &9   &&    ! "  #!$!    <                              &"    /    .     &     *.  .&, .              .& " 9   3                         &      !   "#$%     *     ,  *.  9  ,       &".      &"          &9    &&    &      &      &         &             &"         .     &    *.    &,       ;   "#     "        "        ¼   ¼ &" .  /& 3           )     /<              *:,     %                        '     (       *.  /,   ) & 3     0     &"/ & 3    &    0     &    & 3      0   0   & 0       &&   & 3  0     &  &       &    &  3   0     &    =   * 7     '     (       *.  /,   ) 0     &     &&  0     &  * 3  Æ           "    *.  /,    &       "    *.  /,      &&     &         &                  !        6          &   &           &     &    &        *.  /,      &       *.  /,& >            &    &                 &           ?         &       &    (       0     &               * 3         *.  /,           "          *.  /,   "    *.  /,     &       "    *.  /,    &    3     "             0      &" 0     &        %                  3   *.  /,    &  '%        0  *+, 3            +:   +8 -   0     &           #  @ *8, *1,-   +8  &   0         *2,    +        0     &              * )  @  8+    (       )   ! "  #!$!    A ,     ,  0     &  0     &&  0     & -., ,   0     &&   - *, ,    &         0     & , +    *.  /,  0     &    "    &  * 3      &   &   &   &   &  &   &   &   &   +8           3               5         *8,                 %               & "    .     &   .       .     #           & "        &     &  "              &"         > 0   " 0       "        '      (       *.  /,  ) , 0     0     &  " 0     0     &  " 0     &   & ,    0     &   0       && , 0  0     &  0     &&  0     & , 0  0     &  0     &&  0     & , 0  0     &  &    0     &&  * & B  +8          *.  /,    &  0     &  "    *.  /,    &  0     &  " "    *.  /,      &&      &    !      /.   & %    "    *.  /,    &        0     &&   "      &  &" "         &&     &     0    &&  0       &&  &  &      +8 & B  +8       0     &&  0     &  "     0     &  0     &&     0     &&     )      +:  Æ              "               0     & "     *.  /,  0     &           * 3      *.  /,    &    "    *.  /,  0     &     +8&  @  +1    4             %            7 ++             C 0   " 0       0    0    0     & "     *.  /,  0     &     ;       @  +:            (                    *.  /,  ) & 0      0       0         &&   " 0        &       &    &   : & 0   0   0      &      &   0  0   0      &      &    0     &   "8  0     &   "8  9/ & 0   0   0       &   &             0       &  0       &   &  0         &    "8        0       &&  &  0         &    "8  9/    8&    ! "  #!$!    //      %           "#      /<& B                     *.  /,    &   #(    +           0     &"       &    &&       *.  /,  * *2,  ;                    *.  /,  *.  /,  *.  /,         -  " 0  &              &      !   "#$%           *     ,  *.     &,      &".      &"    /            &     &&     &  *     &  *     &      * 3      /<  ;                 &               D                +                    *.  /,          ) -,   0     &  0     &  " 0     & -, 0   0     &  0     &  " 0     &  * 3       :8                &      E        &        &E        &         &E    3     0   0     &  0     &   0     &       "#    & F  C :+                                  )   0      &  0         &   0      &  " 0     &  /8                     *.  /,                ) 0   0      &     &   &   0      &   &   &  " 0     &     - *, +        *.  /,      0     &  "          * *2,     '         *.  /,  *.  /,         &  0     &   / ) -, +  "*.  /,  *.  /,          -, +  "*.  /,  .  /,         -, +           "  * 3      :8   / ) C  7 ( % /A=+& >  % C  B44   0     A+ /182 8 '  % 12  3   34     25  @ G  D @H  C /AA. + I  B C J% /AA1& B44 K 6 7 1   /A<88= : 4 F ! ) /A=+& *  7  ; 6 1  7 ( % /A=:& >       44  J I4!  KK L 7  8 7 7  ( 7  * . ( D( K /A=8/8 2 ( % /A=8& 12  3   4   4   @ G  D @H  C