# Definition:Euclidean Relation

## Definition

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

### Left-Euclidean

$\mathcal R$ is left-Euclidean if and only if:

$\tuple {x, z} \in \mathcal R \land \tuple {y, z} \in \mathcal R \implies \tuple {x, y} \in \mathcal R$

### Right-Euclidean

$\mathcal R$ is right-Euclidean if and only if:

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

### Euclidean

$\mathcal R$ is Euclidean if and only if it is both left-Euclidean and right-Euclidean.

The concept of a Euclidean relation was named for Euclid.

It derives ultimately from the first of Euclid's common notions.

In the words of Euclid:

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.

## Also see

• Results about Euclidean relations can be found here.