predicates: Relation Predicates

Description

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

Usage

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_endorelation(x)
relation_is_equivalence(x)
relation_is_functional(x)
relation_is_injective(x)
relation_is_irreflexive(x)
relation_is_left_total(x)
relation_is_linear_order(x)
relation_is_partial_order(x)
relation_is_reflexive(x)
relation_is_right_total(x)
relation_is_strict_linear_order(x)
relation_is_strict_partial_order(x)
relation_is_surjective(x)
relation_is_symmetric(x)
relation_is_tournament(x)
relation_is_transitive(x)
relation_is_weak_order(x)
relation_is_preference(x)
relation_is_preorder(x)
relation_is_quasiorder(x)

Arguments

an object inheriting from class relation.

Details

A binary relation is a relation with arity 2. An endorelation $R$ on a set $X$ is a relation with domain $D(R) = (X, X)$, i.e., a binary relation on $X$.

Let us write $x R y$ iff $(x, y)$ is contained in $R$.

A binary relation $R$ is called [object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

An endorelation $R$ is called [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]

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.

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.