trace               package:relations               R Documentation

_T_r_a_c_e_s _o_f _E_n_d_o_r_e_l_a_t_i_o_n_s

_D_e_s_c_r_i_p_t_i_o_n:

     Compute the left or right trace of an endorelation.

_U_s_a_g_e:

     relation_trace(x, which)

_A_r_g_u_m_e_n_t_s:

       x: an endorelation.

   which: one of '"left"' or '"right"', or a unique abbreviation
          thereof.

_D_e_t_a_i_l_s:

     Let R be a crisp endorelation.  The left and right trace of R
     contain all pairs x, y for which z R x implies z R y for all z
     (left trace) or y R z implies x R z for all z (right trace),
     respectively.  These are the largest (in the natural ordering of
     relations with the same domain) relations such that R * S <= R or
     S * R <= R, respectively (where * denotes composition).  In the
     fuzzy case, the memberships of the traces can be defined as the
     infima over the corresponding fuzzy membership implications.  See
     Chapter 2.3 in Fodor and Roubens (1994) for more information.

_R_e_f_e_r_e_n_c_e_s:

     J. Fodor and M. Roubens (1994), _Fuzzy Preference Modelling and
     Multicriteria Decision Support_. Kluwer Academic Publishers:
     Dordrecht.

