Euclidean relation

In mathematics, Euclidean relations are a class of binary relations that formalize "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other."

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: 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:

Right Euclidean property: solid and dashed arrows indicate antecedents and consequents, respectively.
Schematized right Euclidean relation according to property 10. Deeply-colored squares indicate the equivalence classes of R ′ . Pale-colored rectangles indicate possible relationships of elements in X \ran( R ). In these rectangles, relationships may, or may not, hold.