predicates: Relation Predicates

Description

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

Usage

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

Arguments

an object inheriting from class relation.

na.rm

a logical indicating whether tuples with missing memberships are excluded in the predicate computations.

Details

A binary relation is a relation with arity 2.

A relation $R$ on a set $X$ is called homogeneous iff $D(R) = (X, \dots, X)$

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 [object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

A crisp endorelation $R$ is called [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Some combinations of these basic properties have special names because of their widespread use: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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. [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object] The combined predicates are obtained by combining the basic predicates as for crisp endorelations (see above).

A relation has missings iff at least one cell in the incidence matrix is NA.

References

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.