# Definition:Endorelation

## Definition

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 both subclasses 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.

Hence 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:

antireflexive
symmetric
non-transitive.

### 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:

reflexive
antisymmetric
transitive.

### 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:

non-reflexive
non-symmetric
non-transitive.

## Also see

• Results about endorelations can be found here.