nLab idempotent relation

Definition

A binary endorelation $R$ on a set $A$ is idempotent if it is a transitive dense relation

• for all $x \in A$, $y \in A$, and $z \in A$, $R(x, y)$ and $R(y, z)$ implies that $R(x, z)$.

• for all $x \in A$ and $y \in A$ such that $R(x, y)$ there exists an element $z \in A$ such that $R(x, z)$ and $R(z, y)$.

Idempotent relations are idempotent in the sense that in Rel the composition of the relation $R$ with itself is $R$ again, $R \circ R = R$.

Every preorder is a reflexive idempotent relation

Related concepts

• dense relation

• transitive relation

• DLO

• idempotent element

References

See also:

• Wikipedia, Idempotent relation