Homogeneous relation

Template:Short description In mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product Template:Math. This is commonly phrased as "a relation on X" or "a (binary) 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 (undirected) 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 (with arbitrary elements Template:Math, Template:Math) are:

• Empty relation

  Template:Math;
  that is, Template:Math holds never;

• Universal relation

  Template:Math;
  that is, Template:Math holds always;

• Identity relation (see also Identity function)

  Template:Math};
  that is, Template:Math holds if and only if Template:Math.

Example

Fifteen 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

Template:See also Some important properties that a homogeneous relation Template:Mvar over a set Template:Mvar may have are:

Template:Em

for all Template:Math, Template:Math. For example, ≥ is a reflexive relation but > is not.

Template:Em (or Template:Em)

for all Template:Math, not Template:Math. For example, > is an irreflexive relation, but ≥ is not.

Template:Em

for all Template:Math, if Template:Math then Template:Math. 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.

Template:Em

for all Template:Math, if Template:Math then Template:Math.

Template:Em

for all Template:Math, if Template:Math then Template:Math.

Template:Em

for all Template:Math, if Template:Math then Template:Math and Template:Math. 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 Template:Math defined by Template:Math is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair Template:Math, and Template:Math, but not Template:Math, respectively. The latter two facts also rule out (any kind of) quasi-reflexivity.

Template:Em

for all Template:Math, if Template:Math then Template:Math. For example, "is a blood relative of" is a symmetric relation, because Template:Mvar is a blood relative of Template:Mvar if and only if Template:Mvar is a blood relative of Template:Mvar.

Template:Em

for all Template:Math, if Template:Math and Template:Math then Template:Math. For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false).

Template:Em

for all Template:Math, if Template:Math then not Template:Math. 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 Template:Math defined by Template:Math is neither symmetric nor antisymmetric, let alone asymmetric.

Template:Em

for all Template:Math, if Template:Math and Template:Math then Template:Math. 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.

Template:Em

for all Template:Math, if Template:Math and Template:Math then never Template:Math.

Template:Em

if the complement of R is transitive. That is, for all Template:Math, if Template:Math, then Template:Math or Template:Math. This is used in pseudo-orders in constructive mathematics.

Template:Em

for all Template:Math, if Template:Math and Template:Math but neither Template:Math nor Template:Math, then Template:Math but not Template:Math.

Template:Em

for all Template:Math, if Template:Mvar and Template:Mvar are incomparable with respect to Template:Mvar and if the same is true of Template:Mvar and Template:Mvar, then Template:Mvar and Template:Mvar are also incomparable with respect to Template:Mvar. This is used in weak orderings.

Again, the previous 5 alternatives are not exhaustive. For example, the relation Template:Math if (Template:Math or Template:Math) satisfies none of these properties. On the other hand, the empty relation trivially satisfies all of them.

Template:Em

for all Template:Math such that Template:Math, there exists some Template:Math such that Template:Math and Template:Math. This is used in dense orders.

Template:Em

for all Template:Math, if Template:Math then Template:Math or Template:Math. This property is sometimesTemplate:Citation needed called "total", which is distinct from the definitions of "left/right-total" given below.

Template:Em

for all Template:Math, Template:Math or Template:Math. This property, too, is sometimesTemplate:Citation needed called "total", which is distinct from the definitions of "left/right-total" given below.

Template:Em

for all Template:Math, exactly one of Template:Math, Template:Math or Template:Math holds. For example, > is a trichotomous relation on the real numbers, while the relation "divides" over the natural numbers is not.

Template:Em (or just Template:Em)

for all Template:Math, if Template:Math and Template:Math then Template:Math. For example, = is a Euclidean relation because if Template:Math and Template:Math then Template:Math.

Template:Em

for all Template:Math, if Template:Math and Template:Math then Template:Math.

Template:Em

every nonempty subset Template:Mvar of Template:Mvar contains a minimal element with respect to Template:Mvar. Well-foundedness implies the descending chain condition (that is, no infinite chain ... Template:Math 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:

Template:Em

for all Template:Math, the class of all Template:Mvar such that Template:Math is a set. (This makes sense only if relations over proper classes are allowed.)

Template:Em

for all Template:Math and all Template:Math, if Template:Math and Template:Math then Template:Math.

Template:Em

for all Template:Math and all Template:Math, if Template:Math and Template:Math then Template:Math.

Template:Em (also called left-total)

for all Template:Math there exists a Template:Math such that Template:Math. This property is different from the definition of connected (also called total by some authors).Template:Citation needed

Template:Em (also called right-total)

for all Template:Math, there exists an Template:Math such that xRy.

A Template:Em is a relation that is reflexive and transitive. A Template:Em, also called Template:Em or Template:Em, is a relation that is reflexive, transitive, and connected.

A Template:Em, also called Template:Em,Template:Citation needed is a relation that is reflexive, antisymmetric, and transitive. A Template:Em, also called Template:Em,Template:Citation needed is a relation that is irreflexive, antisymmetric, and transitive. A Template:Em, also called Template:Em, Template:Em, or Template:Em, is a relation that is reflexive, antisymmetric, transitive and connected. A Template:Em, also called Template:Em, Template:Em, or Template:Em, is a relation that is irreflexive, antisymmetric, transitive and connected.

A Template:Em is a relation that is symmetric and transitive. An Template:Em 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.

Implications and conflicts between properties of homogeneous binary relations
Implications (blue) and conflicts (red) between properties (yellow) of homogeneous binary relations. For example, every asymmetric relation is irreflexive ("Template:Color Template:Color Template:Color"), and no relation on a non-empty set can be both irreflexive and reflexive ("Template:Color Template:Color Template:Color"). Omitting the red edges results in a Hasse diagram.

Operations

If R is a homogeneous relation over a set X then each of the following is a homogeneous relation over X:

Template:Em, R⁼

Defined as Template:Math 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.

Template:Em, R^≠

Defined as Template:Math} or the largest irreflexive relation over X contained in R.

Template:Em, 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.

Template:Em, R*

Defined as Template:Math, the smallest preorder containing R.

Template:Em, R^≡

Defined as the smallest equivalence relation over X containing R.

All operations defined in Template:Section link also apply to homogeneous relations.

Homogeneous relations by property

	Reflexivity	Symmetry	Transitivity	Connectedness	Symbol	Example
Directed graph					→
Undirected graph		Template:Yes
Dependency	Template:Yes	Template:Yes
Tournament	Template:No	Template:No				Pecking order
Preorder	Template:Yes		Template:Yes		≤	Preference
Total preorder	Template:Yes		Template:Yes	Template:Yes	≤
Partial order	Template:Yes	Template:No	Template:Yes		≤	Subset
Strict partial order	Template:No	Template:No	Template:Yes		<	Strict subset
Total order	Template:Yes	Template:No	Template:Yes	Template:Yes	≤	Alphabetical order
Strict total order	Template:No	Template:No	Template:Yes	Template:Yes	<	Strict alphabetical order
Partial equivalence relation		Template:Yes	Template:Yes
Equivalence relation	Template:Yes	Template:Yes	Template:Yes		~, ≡	Equality

Enumeration

The set of all homogeneous relations $ℬ(X)$ over a set X is the set Template:Math, which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on $ℬ(X)$, it forms a monoid with involution where the identity element is the identity relation.

The number of distinct homogeneous relations over an n-element set is Template:Math Template:OEIS: Template:Number of relations

Notes:

• The number of irreflexive relations is the same as that of reflexive relations.
• The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.
• The number of strict weak orders is the same as that of total preorders.
• The total orders are the partial orders that are also total preorders. The number of preorders that are neither a partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders, minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.
• The number of equivalence relations is the number of partitions, which is the Bell number.

The homogeneous relations can be grouped into pairs (relation, complement), except that for Template:Math the relation is its own complement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse complement).

Examples

• Order relations, including strict orders:
  • Greater than
  • Greater than or equal to
  • Less than
  • Less than or equal to
  • Divides (evenly)
  • Subset of
• Equivalence relations:
  • Equality
  • Parallel with (for affine spaces)
  • Equinumerosity or "is in bijection with"
  • Isomorphic
  • Equipollent line segments
• Tolerance relation, a reflexive and symmetric relation:
  • Dependency relation, a finite tolerance relation
  • Independency relation, the complement of some dependency relation
• Kinship relations

Generalizations

• A binary relation in general need not be homogeneous, it is defined to be a subset Template:Nowrap for arbitrary sets X and Y.
• A finitary relation is a subset Template:Nowrap for some natural number n and arbitrary sets X₁, ..., X_n, it is also called an n-ary relation.