predicates             package:relations             R Documentation

_R_e_l_a_t_i_o_n _P_r_e_d_i_c_a_t_e_s

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

     Predicate functions for testing for binary relations and
     endorelations, and special kinds thereof.

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

     relation_is_Euclidean(x)
     relation_is_Ferrers(x)
     relation_is_antisymmetric(x)
     relation_is_asymmetric(x)
     relation_is_bijective(x)
     relation_is_binary(x)
     relation_is_complete(x)
     relation_is_coreflexive(x)
     relation_is_crisp(x)
     relation_is_endorelation(x)
     relation_is_equivalence(x)
     relation_is_functional(x)
     relation_is_homogeneous(x)
     relation_is_injective(x)
     relation_is_interval_order(x)
     relation_is_irreflexive(x)
     relation_is_left_total(x)
     relation_is_linear_order(x)
     relation_is_match(x)
     relation_is_negatively_transitive(x)
     relation_is_partial_order(x)
     relation_is_preference(x)
     relation_is_preorder(x)
     relation_is_quasiorder(x)
     relation_is_quasitransitive(x)
     relation_is_reflexive(x)
     relation_is_right_total(x)
     relation_is_semiorder(x)
     relation_is_semitransitive(x)
     relation_is_strict_linear_order(x)
     relation_is_strict_partial_order(x)
     relation_is_strongly_complete(x)
     relation_is_surjective(x)
     relation_is_symmetric(x)
     relation_is_tournament(x)
     relation_is_transitive(x)
     relation_is_trichotomous(x)
     relation_is_weak_order(x)

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

       x: an object inheriting from class 'relation'.

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

     A binary relation is a relation with arity 2.

     A relation R on a set X is called _homogeneous_ iff D(R) = (X,
     ..., X)

     An _endorelation_ is a binary homogeneous relation.

     For a crisp binary relation, let us write x R y iff (x, y) is
     contained in R.

     A crisp binary relation R is called

     _l_e_f_t-_t_o_t_a_l: for all x there is at least one y such that x R y.

     _r_i_g_h_t-_t_o_t_a_l: for all y there is at least one x such that x R y.

     _f_u_n_c_t_i_o_n_a_l: for all x there is at most one y such that x R y.

     _s_u_r_j_e_c_t_i_v_e: the same as right-total.

     _i_n_j_e_c_t_i_v_e: for all y there is at most one x such that x R y.

     _b_i_j_e_c_t_i_v_e: left-total, right-total, functional and injective.

     A crisp endorelation R is called

     _r_e_f_l_e_x_i_v_e: x R x for all x.

     _i_r_r_e_f_l_e_x_i_v_e: there is no x such that x R x.

     _c_o_r_e_f_l_e_x_i_v_e: x R y implies x = y.

     _s_y_m_m_e_t_r_i_c: x R y implies y R x.

     _a_s_y_m_m_e_t_r_i_c: x R y implies that not y R x.

     _a_n_t_i_s_y_m_m_e_t_r_i_c: x R y and y R x imply that x = y.

     _t_r_a_n_s_i_t_i_v_e: x R y and y R z imply that x R z.

     _c_o_m_p_l_e_t_e: for all distinct x and y, x R y or y R x.

     _s_t_r_o_n_g_l_y _c_o_m_p_l_e_t_e for all x and y, x R y or y R x (i.e., complete
          and reflexive).

     _n_e_g_a_t_i_v_e_l_y _t_r_a_n_s_i_t_i_v_e not x R y and not y R z imply that not x R
          z.

     _F_e_r_r_e_r_s x R y and z R w imply x R w or y R z.

     _s_e_m_i_t_r_a_n_s_i_t_i_v_e x R y and y R z imply x R w or w R z.

     _q_u_a_s_i_t_r_a_n_s_i_t_i_v_e x R y and not y R x and y R z and not z R y imply
          x R z and not z R x (i.e., the asymmetric part of R is
          transitive).

     _t_r_i_c_h_o_t_o_m_o_u_s exactly one of x R y, y R x, or x = y holds.

     _E_u_c_l_i_d_e_a_n x R y and x R z imply y R z.

     Some combinations of these basic properties have special names
     because of their widespread use:

     _p_r_e_o_r_d_e_r: reflexive and transitive.

     _q_u_a_s_i_o_r_d_e_r: the same as preorder.

     _e_q_u_i_v_a_l_e_n_c_e: a symmetric preorder (reflexive, symmetric, and
          transitive).

     _w_e_a_k _o_r_d_e_r: a complete preorder (complete, reflexive, and
          transitive).

     _p_r_e_f_e_r_e_n_c_e: the same as weak order.

     _p_a_r_t_i_a_l _o_r_d_e_r: an antisymmetric preorder (reflexive,
          antisymmetric, and transitive).

     _s_t_r_i_c_t _p_a_r_t_i_a_l _o_r_d_e_r: irreflexive, antisymmetric, and transitive,
          or equivalently: asymmetric and transitive).

     _l_i_n_e_a_r _o_r_d_e_r: a complete partial order.

     _s_t_r_i_c_t _l_i_n_e_a_r _o_r_d_e_r: a complete strict partial order.

     _m_a_t_c_h: strongly complete.

     _t_o_u_r_n_a_m_e_n_t: complete and asymmetric.

     _i_n_t_e_r_v_a_l _o_r_d_e_r: complete and Ferrers.

     _s_e_m_i_o_r_d_e_r: a semitransitive interval order.

     If R is a weak order ((weak) preference relation), I = I(R)
     defined by x I y iff x R y and y R x is an equivalence, the
     _indifference relation_ corresponding to R.

     There seem to be no commonly agreed definitions for order
     relations: e.g., Fishburn (1972) requires these to be irreflexive.

     For a fuzzy binary relation R, let R(x, y) denote the membership
     of (x, y) in the relation.  Write T and S for the fuzzy t-norm
     (intersection) and t-conorm (disjunction), respectively (min and
     max for the standard Zadeh family). Then generalizations of the
     above basic endorelation predicates are as follows.

     _r_e_f_l_e_x_i_v_e: R(x, x) = 1 for all x.

     _i_r_r_e_f_l_e_x_i_v_e: R(x, x) = 0 for all x.

     _c_o_r_e_f_l_e_x_i_v_e: R(x, y) > 0 implies x = y.

     _s_y_m_m_e_t_r_i_c: R(x, y) = R(y, x) for all x, y.

     _a_s_y_m_m_e_t_r_i_c: T(R(x, y), R(y, x)) = 0 for all x, y. 

     _a_n_t_i_s_y_m_m_e_t_r_i_c: T(R(x, y), R(y, x)) = 0 for all x != y.

     _t_r_a_n_s_i_t_i_v_e: T(R(x, y), R(y, z)) <= R(x, z) for all x, y, z.

     _c_o_m_p_l_e_t_e: S(R(x, y), R(y, x)) = 1 for all x != y.

     _s_t_r_o_n_g_l_y _c_o_m_p_l_e_t_e: S(R(x, y), R(y, x)) = 1 for all x, y.

     _n_e_g_a_t_i_v_e_l_y _t_r_a_n_s_i_t_i_v_e: R(x, z) <= S(R(x, y), R(y, z)) for all x,
          y, z.

     _F_e_r_r_e_r_s: T(R(x, y), R(z, w)) <= S(R(x, w), R(z, y)) for all x, y,
          z, w.

     _s_e_m_i_t_r_a_n_s_i_t_i_v_e: T(R(x, w), R(w, y)) <= S(R(x, z), R(z, y)) for all
          x, y, z, w.

     The combined predicates are obtained by combining the basic
     predicates as for crisp endorelations (see above).

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

     P. C. Fishburn (1972), _Mathematics of decision theory_. Methods
     and Models in the Social Sciences 3. Mouton: The Hague.

     H. R. Varian (2002), _Intermediate Microeconomics: A Modern
     Approach_. 6th Edition. W. W. Norton & Company.

