Definition:Endorelation/Class Theory

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Definition

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 $A$.


Notation

If $\tuple {x, y}$ is an ordered pair such that $\tuple {x, y} \in \RR$, we use the notation:

$s \mathrel \RR t$

or:

$\map \RR {s, t}$

and can say:

$s$ bears $\RR$ to $t$
$s$ stands in the relation $\RR$ to $t$


If $\tuple {s, t} \notin \RR$, we can write: $s \not \mathrel \RR t$, that is, by drawing a line through the relation symbol.


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.


Also see

  • Results about relations can be found here.


Sources