Definition:Euclidean Relation

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


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

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


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

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


$\mathcal R$ 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.