Homogeneous relation
In mathematics, a homogeneous relation on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product. This is commonly phrased as "a relation on X" or "a relation over X". An example of a homogeneous relation is the relation of kinship, where the relation is between people.
Common types of endorelations include orders, graphs, and equivalences. Specialized studies of order theory and graph theory have developed understanding of endorelations. Terminology particular for graph theory is used for description, with an ordinary graph presumed to correspond to a symmetric relation, and a general endorelation corresponding to a directed graph. An endorelation R corresponds to a logical matrix of 0s and 1s, where the expression xRy corresponds to an edge between x and y in the graph, and to a 1 in the square matrix of R. It is called an adjacency matrix in graph terminology.
Particular homogeneous relations
Some particular homogeneous relations over a set X are:;Empty relation
;Universal relation
;Identity relation
Example
| Af | An | Ar | Au | Ca | Co | Eu | In | Ju | NA | Na | Pa | Ph | SA | Sc | So | ||
| African | Af | ||||||||||||||||
| Antarctic | An | ||||||||||||||||
| Arabian | Ar | ||||||||||||||||
| Australian | Au | ||||||||||||||||
| Caribbean | Ca | ||||||||||||||||
| Cocos | Co | ||||||||||||||||
| Eurasian | Eu | ||||||||||||||||
| Indian | In | ||||||||||||||||
| Juan de Fuca | Ju | ||||||||||||||||
| North american | NA | ||||||||||||||||
| Nazca | Na | ||||||||||||||||
| Pacific | Pa | ||||||||||||||||
| Philippine | Ph | ||||||||||||||||
| South american | SA | ||||||||||||||||
| Scotia | Sc | ||||||||||||||||
| Somali | So |
File:Tectonic plates.svg|thumb|right|The binary relation that describes whether two tectonic plates are in contact is a homogenous relation, because both the first and second argument are from the same set, that is the set of tectonic plates on Earth.
Sixteen large tectonic plates of the Earth's crust contact each other in a homogeneous relation. The relation can be expressed as a logical matrix with 1 indicating contact and 0 no contact. This example expresses a symmetric relation.
Properties
Some important properties that a homogeneous relation over a set may have are:; : for all,. For example, ≥ is a reflexive relation but > is not.
; : 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 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.
; : 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 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.
; : 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. 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.
; : for all and all, if and then.
; : for all and all, if and then.
; : for all there exists a such that. This property is different from the definition of connected.
; : 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 is one whose inverse is univalent. A is one that is right-total.
| Implications and conflicts between properties of homogeneous binary relations |
Operations
If R is a homogeneous relation over a set X then each of the following is a homogeneous relation over X:;, R= : Defined as or the smallest reflexive relation over X containing R. This can be proven to be equal to the intersection of all reflexive relations containing R.
;, R≠ : Defined as or the largest irreflexive relation over X contained in R.
;, R+ : Defined as the smallest transitive relation over X containing R. This can be seen to be equal to the intersection of all transitive relations containing R.
;, R* : Defined as, the smallest preorder containing R.
;, R≡ : Defined as the smallest equivalence relation over X containing R.
All operations defined in also apply to homogeneous relations.