# Definition:Endorelation

## Definition

Let $S \times S$ be the cartesian product of a set or class $S$ with itself.

Let $\RR$ be a relation on $S \times S$.

Then $\RR$ is referred to as an **endorelation on $S$**.

An **endorelation** can be defined as the ordered triple:

- $\RR = \tuple {S, S, R}$

where $R \subseteq S \times S$.

### General Definition

An $n$-ary relation $\RR$ on a cartesian space $S^n$ is an **$n$-ary endorelation on $S$**:

- $\RR = \struct {S, S, \ldots, S, R}$

where $R \subseteq S^n$.

### Class Theoretical Definition

In the context of class theory, the definition follows the same lines:

Let $A$ be a class.

An **endorelation** $\RR$ on $A$ is a subclass of the Cartesian product $A \times A$.

That is, such that the domain and image of $\RR$ are bothsubclasses of $A$.

## Also known as

The term **endorelation** is rarely seen. Once it is established that the domain and codomain of a given relation are the same, further comment is rarely needed.

An **endorelation on $S$** is also called a **relation in $S$**, or a **relation on $S$**. The latter term is discouraged, though, because it can also mean a **left-total relation**, and confusion can arise.

Some sources use the term **binary relation** exclusively to refer to a **binary endorelation**.

Some sources, for example 1974: P.M. Cohn: *Algebra: Volume $\text { 1 }$*, use the term **relation** for what is defined here as an **endorelation**, and a **relation** defined as a general ordered triple of sets: $\tuple {S, T, R \subseteq S \times T}$ is called a **correspondence**.

As this can cause confusion with the usage of correspondence to mean a relation which is both left-total and right-total, it is recommended that this is not used.

## Examples

### Arbitrary Relation

Let $V_0 = \set {a, b, c}$.

A possible endorelation on $V_0$ is:

- $R = \set {\tuple {a, a}, \tuple {b, b}, \tuple {b, c}, \tuple {c, b} }$

### Properties of Arbitrary Relation: 1

Let $V = \set {u, v, w, x}$.

Let $E$ be the relation on $V$ defined as:

- $E = \set {\tuple {u, v}, \tuple {v, u}, \tuple {v, w}, \tuple {w, v} }$

Then $E$ is:

### Properties of Arbitrary Relation: 2

Let $V = \set {a, b, c, d}$.

Let $R$ be the relation on $V$ defined as:

- $E = \set {\tuple {a, a}, \tuple {a, b}, \tuple {a, c}, \tuple {a, d}, \tuple {b, b}, \tuple {b, c}, \tuple {b, d}, \tuple {c, c}, \tuple {c, d}, \tuple {d, d} }$

Then $E$ is:

### Properties of Arbitrary Relation: 3

Let $V = \set {a, b, c, d}$.

Let $R$ be the relation on $V$ defined as:

- $r = \set {\tuple {a, a}, \tuple {a, b}, \tuple {b, b}, \tuple {b, c}, \tuple {c, b}, \tuple {c, c} }$

Then $E$ is:

## Also see

- Results about
**endorelations**can be found here.

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 3$: Equivalence relations - 1959: E.M. Patterson:
*Topology*(2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 10$. Equivalence Relations - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.2$: Definition $1.5$ - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 2.1$. Relations on a set - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 10$: Equivalence Relations - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.3$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Relations - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 1.4$: Decomposition of a set into classes. Equivalence relations - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations: Definition $19.1$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 7$: Relations - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.4$: Equivalence relations - 1993: Keith Devlin:
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*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations