Euclidean relation

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In mathematics, a binary relation R on a set X is Euclidean if it satisfies the following: for every a, b, c in X, if a is related to b and c, then b is related to c.

To write this in predicate logic:

This property is different from transitivity. However, if a relation is reflexive and symmetric, then it is Euclidean if and only if it is transitive.

If a relation is Euclidean and reflexive, it is also symmetric and transitive, hence it is an equivalence relation. Consequently, equivalence relations are exactly the reflexive Euclidean relations.