predicates {relations} | R Documentation |
Predicate functions for testing for binary relations and endorelations, and special kinds thereof.
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)
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an object inheriting from class relation . |
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
A crisp endorelation R is called
Some combinations of these basic properties have special names because of their widespread use:
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.
The combined predicates are obtained by combining the basic predicates as for crisp endorelations (see above).
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.