Some important properties that a homogeneous relation over a set may have are: ; : for all , . For example, ≥ is a reflexive relation but > is not. ; (or ) : for all , not . For example, > is an irreflexive relation, but ≥ is not. ; : for all , if then . For example, the relation over the integers in which each odd number is related to itself is a coreflexive relation. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation. ; : for all , if then . ; : for all , if then . ; : for all , if then and . A relation is quasi-reflexive if, and only if, it is both left and right quasi-reflexive. The previous 6 alternatives are far from being exhaustive; e.g., the binary relation defined by is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair , and , but not , respectively. The latter two facts also rule out (any kind of) quasi-reflexivity. ; : for all , if then . For example, "is a blood relative of" is a symmetric relation, because is a blood relative of if and only if is a blood relative of . ; : for all , if and then . For example, ≥ is an antisymmetric relation; so is >, but
vacuously (the condition in the definition is always false). ; : for all , if then not . A relation is asymmetric if and only if it is both antisymmetric and irreflexive. For example, > is an asymmetric relation, but ≥ is not. Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation defined by is neither symmetric nor antisymmetric, let alone asymmetric. ; : for all , if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. ; : for all , if and then never . ; : if the complement of
R is transitive. That is, for all , if , then or . This is used in
pseudo-orders in constructive mathematics. ; : for all , if and but neither nor , then but not . ; : for all , if and are incomparable with respect to and if the same is true of and , then and are also incomparable with respect to . This is used in
weak orderings. Again, the previous 5 alternatives are not exhaustive. For example, the relation if ( or ) satisfies none of these properties. On the other hand, the empty relation trivially satisfies all of them. ; : for all such that , there exists some such that and . This is used in
dense orders. ; : for all , if then or . This property is sometimes called "total", which is distinct from the definitions of "left/right-total" given below. ; : for all , or . This property, too, is sometimes called "total", which is distinct from the definitions of "left/right-total" given below. ; : for all , exactly one of , or holds. For example, > is a trichotomous relation on the real numbers, while the relation "divides" over the natural numbers is not. ; (or just ) : for all , if and then . For example, = is a Euclidean relation because if and then . ; : for all , if and then . ; : every nonempty subset of contains a
minimal element with respect to . Well-foundedness implies the
descending chain condition (that is, no infinite chain can exist). If the
axiom of dependent choice is assumed, both conditions are equivalent. Moreover, all properties of binary relations in general also may apply to homogeneous relations: ; : for all , the
class of all such that is a set. (This makes sense only if relations over proper classes are allowed.) ; : for all and all , if and then . ; : for all and all , if and then . ; (also called left-total) : for all there exists a such that . This property is different from the definition of
connected (also called
total by some authors). ; (also called right-total) : for all , there exists an such that
xRy. A is a relation that is reflexive and transitive. A , also called or , is a relation that is reflexive, transitive, and connected. A , also called , is a relation that is reflexive, antisymmetric, and transitive. A , also called , is a relation that is irreflexive, antisymmetric, and transitive. A , also called , , or , is a relation that is reflexive, antisymmetric, transitive and connected. A , also called , , or , is a relation that is irreflexive, antisymmetric, transitive and connected. A is a relation that is symmetric and transitive. An is a relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and total, since these properties imply reflexivity. A univalent relation may also be called a . A is a partial function that is left-total. An (or partial function) is one whose inverse is univalent. A is one that is right-total. == Operations ==