In mathematics, Euclidean relations are a class of binary relations that satisfy a modified form of transitivity that formalizes Euclid's "Common Notion 1" in The Elements: things which equal the same thing also equal one another.
To write this in predicate logic:
Dually, a relation R on X is left Euclidean if for every a, b, c in X, if b is related to a and c is related to a, then b is related to c:
Relation to transitivity
The property of being Euclidean is different from transitivity. A transitive relation is Euclidean only if it is also symmetric. Only a symmetric Euclidean relation is transitive.