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 $X \times X$ Script error: No such module "Check for unknown parameters"..^[1]^[2]^[3] This is commonly phrased as "a relation on X"^[4] or "a (binary) relation over X".^[5]^[6] 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 (x is R-related to y) 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 $x 1$ Script error: No such module "Check for unknown parameters"., $x 2$ Script error: No such module "Check for unknown parameters".) are:

Empty relation
$E = \emptyset$ Script error: No such module "Check for unknown parameters".;
that is, $x 1 Ex 2$ Script error: No such module "Check for unknown parameters". holds never;
Universal relation
$U = X \times X$ Script error: No such module "Check for unknown parameters".;
that is, $x 1 Ux 2$ Script error: No such module "Check for unknown parameters". holds always;
Identity relation (see also Identity function)
$I = {(x, x) | x \in X$ Script error: No such module "Check for unknown parameters".};
that is, $x 1 Ix 2$ Script error: No such module "Check for unknown parameters". holds if and only if $x 1 = x 2$ Script error: No such module "Check for unknown parameters"..

Example

Matrix representation of the relation "is adjacent to" on the set of tectonic plates
		Af	An	Ar	Au	Ca	Co	Eu	In	Ju	NA	Na	Pa	Ph	SA	Sc	So
African	Af	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Ya
Antarctic	An	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Ya	Template:Ya
Arabian	Ar	Template:Ya	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya
Australian	Au	Template:Na	Template:Ya	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya
Caribbean	Ca	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Na
Cocos	Co	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na
Eurasian	Eu	Template:Ya	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Na
Indian	In	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya
Juan de Fuca	Ju	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na
North american	NA	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na
Nazca	Na	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Na	Template:Na
Pacific	Pa	Template:Na	Template:Ya	Template:Na	Template:Ya	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na
Philippine	Ph	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na
South american	SA	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na
Scotia	Sc	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na
Somali	So	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya

File:Tectonic plates (2022).svg

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 (depicted "File:Green check.svg") indicating contact and 0 ("File:Dark Red x.svg") no contact. This example expresses a symmetric relation.

Properties

Script error: No such module "Labelled list hatnote". Some important properties that a homogeneous relation Template:Mvar over a set Template:Mvar may have are:

Template:Em: for all $x \in X$ Script error: No such module "Check for unknown parameters"., $xRx$ Script error: No such module "Check for unknown parameters".. For example, ≥ is a reflexive relation but > is not.
Template:Em (or Template:Em): for all $x \in X$ Script error: No such module "Check for unknown parameters"., not $xRx$ Script error: No such module "Check for unknown parameters".. For example, > is an irreflexive relation, but ≥ is not.
Template:Em: for all $x, y \in X$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". then $x = y$ Script error: No such module "Check for unknown parameters"..^[7] 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 $x, y \in X$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". then $xRx$ Script error: No such module "Check for unknown parameters"..
Template:Em: for all $x, y \in X$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". then $yRy$ Script error: No such module "Check for unknown parameters"..
Template:Em: for all $x, y \in X$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". then $xRx$ Script error: No such module "Check for unknown parameters". and $yRy$ Script error: No such module "Check for unknown parameters".. 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 $xRy$ Script error: No such module "Check for unknown parameters". defined by $y = x 2$ Script error: No such module "Check for unknown parameters". is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair $(0, 0)$ Script error: No such module "Check for unknown parameters"., and $(2, 4)$ Script error: No such module "Check for unknown parameters"., but not $(2, 2)$ Script error: No such module "Check for unknown parameters"., respectively. The latter two facts also rule out (any kind of) quasi-reflexivity.

Template:Em: for all $x, y \in X$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". then $yRx$ Script error: No such module "Check for unknown parameters".. 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 $x, y \in X$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". and $yRx$ Script error: No such module "Check for unknown parameters". then $x = y$ Script error: No such module "Check for unknown parameters".. For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false).^[8]
Template:Em: for all $x, y \in X$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". then not $yRx$ Script error: No such module "Check for unknown parameters".. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.^[9] 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 $xRy$ Script error: No such module "Check for unknown parameters". defined by $x > 2$ Script error: No such module "Check for unknown parameters". is neither symmetric nor antisymmetric, let alone asymmetric.

Template:Em: for all $x, y, z \in X$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". and $yRz$ Script error: No such module "Check for unknown parameters". then $xRz$ Script error: No such module "Check for unknown parameters".. A transitive relation is irreflexive if and only if it is asymmetric.^[10] For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
Template:Em: for all $x, y, z \in X$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". and $yRz$ Script error: No such module "Check for unknown parameters". then never $xRz$ Script error: No such module "Check for unknown parameters"..
Template:Em: if the complement of R is transitive. That is, for all $x, y, z \in X$ Script error: No such module "Check for unknown parameters"., if $xRz$ Script error: No such module "Check for unknown parameters"., then $xRy$ Script error: No such module "Check for unknown parameters". or $yRz$ Script error: No such module "Check for unknown parameters".. This is used in pseudo-orders in constructive mathematics.
Template:Em: for all $x, y, z \in X$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". and $yRz$ Script error: No such module "Check for unknown parameters". but neither $yRx$ Script error: No such module "Check for unknown parameters". nor $zRy$ Script error: No such module "Check for unknown parameters"., then $xRz$ Script error: No such module "Check for unknown parameters". but not $zRx$ Script error: No such module "Check for unknown parameters"..
Template:Em: for all $x, y, z \in X$ Script error: No such module "Check for unknown parameters"., 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 $xRy$ Script error: No such module "Check for unknown parameters". if ( $y = 0$ Script error: No such module "Check for unknown parameters". or $y = x +1$ Script error: No such module "Check for unknown parameters".) satisfies none of these properties. On the other hand, the empty relation trivially satisfies all of them.

Template:Em: for all $x, y \in X$ Script error: No such module "Check for unknown parameters". such that $xRy$ Script error: No such module "Check for unknown parameters"., there exists some $z \in X$ Script error: No such module "Check for unknown parameters". such that $xRz$ Script error: No such module "Check for unknown parameters". and $zRy$ Script error: No such module "Check for unknown parameters".. This is used in dense orders.
Template:Em: for all $x, y \in X$ Script error: No such module "Check for unknown parameters"., if $x \neq y$ Script error: No such module "Check for unknown parameters". then $xRy$ Script error: No such module "Check for unknown parameters". or $yRx$ Script error: No such module "Check for unknown parameters".. This property is sometimesScript error: No such module "Unsubst". called "total", which is distinct from the definitions of "left/right-total" given below.
Template:Em: for all $x, y \in X$ Script error: No such module "Check for unknown parameters"., $xRy$ Script error: No such module "Check for unknown parameters". or $yRx$ Script error: No such module "Check for unknown parameters".. This property, too, is sometimesScript error: No such module "Unsubst". called "total", which is distinct from the definitions of "left/right-total" given below.
Template:Em: for all $x, y \in X$ Script error: No such module "Check for unknown parameters"., exactly one of $xRy$ Script error: No such module "Check for unknown parameters"., $yRx$ Script error: No such module "Check for unknown parameters". or $x = y$ Script error: No such module "Check for unknown parameters". holds. For example, > is a trichotomous relation on the real numbers, while the relation "divides" over the natural numbers is not.^[11]
Template:Em (or just Template:Em): for all $x, y, z \in X$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". and $xRz$ Script error: No such module "Check for unknown parameters". then $yRz$ Script error: No such module "Check for unknown parameters".. For example, = is a Euclidean relation because if $x = y$ Script error: No such module "Check for unknown parameters". and $x = z$ Script error: No such module "Check for unknown parameters". then $y = z$ Script error: No such module "Check for unknown parameters"..
Template:Em: for all $x, y, z \in X$ Script error: No such module "Check for unknown parameters"., if $yRx$ Script error: No such module "Check for unknown parameters". and $zRx$ Script error: No such module "Check for unknown parameters". then $yRz$ Script error: No such module "Check for unknown parameters"..
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 ... $x n R ... Rx 3 Rx 2 Rx 1$ Script error: No such module "Check for unknown parameters". can exist). If the axiom of dependent choice is assumed, both conditions are equivalent.^[12]^[13]

Moreover, all properties of binary relations in general also may apply to homogeneous relations:

Template:Em: for all $x \in X$ Script error: No such module "Check for unknown parameters"., the class of all Template:Mvar such that $yRx$ Script error: No such module "Check for unknown parameters". is a set. (This makes sense only if relations over proper classes are allowed.)
Template:Em: for all $x, z \in X$ Script error: No such module "Check for unknown parameters". and all $y \in Y$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". and $zRy$ Script error: No such module "Check for unknown parameters". then $x = z$ Script error: No such module "Check for unknown parameters"..
Template:Em: for all $x \in X$ Script error: No such module "Check for unknown parameters". and all $y, z \in Y$ Script error: No such module "Check for unknown parameters"., if $xRy$ Script error: No such module "Check for unknown parameters". and $xRz$ Script error: No such module "Check for unknown parameters". then $y = z$ Script error: No such module "Check for unknown parameters"..^[14]
Template:Em (also called left-total): for all $x \in X$ Script error: No such module "Check for unknown parameters". there exists a $y \in Y$ Script error: No such module "Check for unknown parameters". such that $xRy$ Script error: No such module "Check for unknown parameters".. This property is different from the definition of connected (also called total by some authors).Script error: No such module "Unsubst".
Template:Em (also called right-total): for all $y \in Y$ Script error: No such module "Check for unknown parameters"., there exists an $x \in X$ Script error: No such module "Check for unknown parameters". 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,Script error: No such module "Unsubst". is a relation that is reflexive, antisymmetric, and transitive. A Template:Em, also called Template:Em,Script error: No such module "Unsubst". 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.^[15] 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
File:BinRelProp Impl Confl.gif Implications (blue) and conflicts (red) between properties (yellow) of homogeneous binary relations. For example, every asymmetric relation is irreflexive ("<templatestyles src="Template:Color/styles.css" />ASym <templatestyles src="Template:Color/styles.css" />⇒ <templatestyles src="Template:Color/styles.css" />Irrefl"), and no relation on a non-empty set can be both irreflexive and reflexive ("<templatestyles src="Template:Color/styles.css" />Irrefl <templatestyles src="Template:Color/styles.css" /># <templatestyles src="Template:Color/styles.css" />Refl"). 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 $R = = {(x, x) | x \in X} \cup R$ Script error: No such module "Check for unknown parameters". 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 $R≠ = R \ {(x, x) | x ∈ X$ Script error: No such module "Check for unknown parameters".} 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 $R * = (R +) =$ Script error: No such module "Check for unknown parameters"., 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		Symmetric
Dependency	Reflexive	Symmetric
Tournament	Irreflexive	Asymmetric				Pecking order
Preorder	Reflexive		Transitive		≤	Preference
Total preorder	Reflexive		Transitive	Connected	≤
Partial order	Reflexive	Antisymmetric	Transitive		≤	Subset
Strict partial order	Irreflexive	Asymmetric	Transitive		<	Strict subset
Total order	Reflexive	Antisymmetric	Transitive	Connected	≤	Alphabetical order
Strict total order	Irreflexive	Asymmetric	Transitive	Connected	<	Strict alphabetical order
Partial equivalence relation		Symmetric	Transitive
Equivalence relation	Reflexive	Symmetric	Transitive		~, ≡	Equality

Enumeration

The set of all homogeneous relations $ℬ (X)$ over a set X is the set $2 X \times X$ Script error: No such module "Check for unknown parameters"., 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.^[16]

The number of distinct homogeneous relations over an n-element set is $2 n 2$ Script error: No such module "Check for unknown parameters". (sequence A002416 in the OEIS):

Number of n-element binary relations of different types
ElemTemplate:Soft hyphenents	Any	Transitive	Reflexive	Symmetric	Preorder	Partial order	Total preorder	Total order	Equivalence relation
0	1	1	1	1	1	1	1	1	1
1	2	2	1	2	1	1	1	1	1
2	16	13	4	8	4	3	3	2	2
3	512	171	64	64	29	19	13	6	5
4	Script error: No such module "val".	Script error: No such module "val".	Script error: No such module "val".	Script error: No such module "val".	355	219	75	24	15
n	2^n²		2ⁿ⁽ⁿ⁻¹⁾	2^n(n+1)/2			∑Script error: No such module "Su". k!S(n, k)	n!	∑Script error: No such module "Su". S(n, k)
OEIS	A002416	A006905	A053763	A006125	A000798	A001035	A000670	A000142	A000110

Note that S(n, k) refers to Stirling numbers of the second kind.

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 $n = 0$ Script error: No such module "Check for unknown parameters". 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 R ⊆ X × Y for arbitrary sets X and Y.
A finitary relation is a subset R ⊆ X₁ × ... × X_n for some natural number n and arbitrary sets X₁, ..., X_n, it is also called an n-ary relation.

References

↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Fonseca de Oliveira, J. N., & Pereira Cunha Rodrigues, C. D. J. (2004). Transposing Relations: From Maybe Functions to Hash Tables. In Mathematics of Program Construction (p. 337).
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1"..
↑ Script error: No such module "citation/CS1". Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".
↑ Since neither 5 divides 3, nor 3 divides 5, nor 3=5.
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Gunther Schmidt & Thomas Strohlein (2012)[1987] Template:Trim&pg=PA54 Relations and Graphs, p. 54, at Google Books
↑ Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, Template:ISBN, p. 4
↑ Script error: No such module "citation/CS1".

Script error: No such module "Check for unknown parameters".

[Winter2007-1] Script error: No such module "citation/CS1".

[Müller2012-2] Script error: No such module "citation/CS1".

[PahlDamrath2001-p496-3] Script error: No such module "citation/CS1".

[4] Script error: No such module "citation/CS1".

[5] Script error: No such module "citation/CS1".

[6] Script error: No such module "citation/CS1".

[7] Fonseca de Oliveira, J. N., & Pereira Cunha Rodrigues, C. D. J. (2004). Transposing Relations: From Maybe Functions to Hash Tables. In Mathematics of Program Construction (p. 337).

[8] Script error: No such module "citation/CS1".

[9] Script error: No such module "citation/CS1"..

[10] Script error: No such module "citation/CS1". Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".

[11] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[12] Script error: No such module "citation/CS1".

[13] Script error: No such module "citation/CS1".

[14] Gunther Schmidt & Thomas Strohlein (2012)[1987] Template:Trim&pg=PA54 Relations and Graphs, p. 54, at Google Books

[15] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, Template:ISBN, p. 4

[16] Script error: No such module "citation/CS1".

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Homogeneous relation

Contents

Particular homogeneous relations

Example

Properties

Operations

Enumeration

Examples

Generalizations

References

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Homogeneous relation

Particular homogeneous relations

Example

Properties

Operations

Enumeration

Examples

Generalizations

References

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