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Concept# Homogeneous relation

Summary

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 × X. 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.
Some particular homogeneous relations over a set X (with arbitrary elements x_1, x_2) are:
Empty relation
E = ∅; that is, x_1Ex_2 holds never;
Universal relation
U = X × X; that is, x_1Ux_2 holds always;
Identity relation (see also identity function)
I = {(x, x) x ∈ X}; that is, x_1Ix_2 holds if and only if x_1 = x_2.
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.
Category:Binary relations
Some important properties that a homogeneous relation R over a set X may have are:
for all x ∈ X, xRx. For example, ≥ is a reflexive relation but > is not.
(or ) for all x ∈ X, not xRx. For example, > is an irreflexive relation, but ≥ is not.
for all x, y ∈ X, if xRy then x = y. 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.

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Euclidean relation

In mathematics, Euclidean relations are a class of binary relations that formalize "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other." A binary relation R on a set X is Euclidean (sometimes called right Euclidean) if it satisfies the following: for every a, b, c in X, if a is related to b and c, then b is related to c.

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In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation. Bertrand Russell used serial relations in The Principles of Mathematics (1903) as he explored the foundations of order theory and its applications. The term serial relation was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness, irreflexivity, and transitivity.

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