Definition:Euclidean Relation

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

Left-Euclidean
$$\mathcal{R}$$ is Left-Euclidean iff:


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

Right-Euclidean
$$\mathcal{R}$$ is Right-Euclidean iff:


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

Euclidean
$$\mathcal{R}$$ is Euclidean iff:


 * $$\mathcal{R}$$ is left-Euclidean;
 * $$\mathcal{R}$$ is right-Euclidean.

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.