Definition:Endorelation/Class Theory
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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 8$ Relations