# Definition:Euclidean Relation

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## 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.*

(*The Elements*: Book $\text{I}$: Common Notions: Common Notion $1$)

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.