Euclidean relation

In mathematics, Euclidean relations are a class of binary relations 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.[1]

To write this in predicate logic:

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

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:

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

Relation to transitivity

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

A relation which is both Euclidean and reflexive is also symmetric and therefore an equivalence relation.[1]