Euclidean relation

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.

Definition

A binary relation R on a set X is Euclidean (sometimes called right 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:

$\forall a, b, c\in X\,(a\,R\, b \land a \,R\, c \to b \,R\, c).$

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:

$\forall a, b, c\in X\,(b\,R\, a \land c \,R\, a \to b \,R\, c).$

Relation to transitivity

The property of being Euclidean is different from transitivity. Only if a transitive relation is also symmetric then it is Euclidean. Only a symmetric Euclidean relation is transitive.

A relation which is both Euclidean and reflexive is also symmetric and therefore an equivalence relation.