Definition:Endorelation

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Definition

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

Let $\mathcal R$ be a relation on $S \times S$.


Then $\mathcal R$ is referred to as an endorelation on $S$.


An endorelation can be defined as the ordered triple:

$\mathcal R = \tuple {S, S, R}$

where $R \subseteq S \times S$.


General Definition

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

$\mathcal R = \left({S, S, \ldots, S, R}\right)$

where $R \subseteq S^n$.


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 set, further comment is rarely needed.

An endorelation 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 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.


Sources