Definition:Euclidean Relation

Definition

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

Left-Euclidean

$\mathcal R$ is Left-Euclidean iff:

$\left({x, z}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, y}\right) \in \mathcal R$

Right-Euclidean

$\mathcal R$ is Right-Euclidean iff:

$\left({x, y}\right) \in \mathcal R \land \left({x, z}\right) \in \mathcal R \implies \left({y, z}\right) \in \mathcal R$

Euclidean

$\mathcal R$ is Euclidean iff:

• $\mathcal R$ is left-Euclidean
• $\mathcal R$ is right-Euclidean.

Also see

• Results about Euclidean relations can be found here.

Source of Name

This entry was named for Euclid.

It derives ultimately from the first of Euclid's Common Notions:

Things which are equal to the same thing are also equal to each other.

However, Euclid did not delve deeply into the field of relation theory.

The concept of equivalence relations was a much later development.