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*:

- Due to the commutativity of ∧ in the definition's antecedent,
*aRb*∧*aRc*even implies*bRc*∧*cRb*when*R*is right Euclidean. Similarly,*bRa*∧*cRa*implies*bRc*∧*cRb*when*R*is left Euclidean. - The property of being Euclidean is different from transitivity. For example, ≤ is transitive, but not right Euclidean,
^{[2]}while*xRy*defined by 0 ≤*x*≤*y*+ 1 ≤ 2 is not transitive,^{[3]}but right Euclidean on natural numbers. - For symmetric relations, transitivity, right Euclideanness, and left Euclideanness all coincide. However, a non-symmetric relation can also be both transitive and right Euclidean, for example,
*xRy*defined by*y*=0. - A relation that is both right Euclidean and reflexive is also symmetric and therefore an equivalence relation.
^{[1]}^{[4]}Similarly, each left Euclidean and reflexive relation is an equivalence. - The range of a right Euclidean relation is always a subset
^{[5]}of its domain. The restriction of a right Euclidean relation to its range is always reflexive,^{[6]}and therefore an equivalence. Similarly, the domain of a left Euclidean relation is a subset of its range, and the restriction of a left Euclidean relation to its domain is an equivalence. Therefore, a right Euclidean relation on*X*that is also right total (respectively a left Euclidean relation on*X*that is also left total) is an equivalence, since its range (respectively its domain) is*X*.^{[7]} - A relation
*R*is both left and right Euclidean, if, and only if, the domain and the range set of*R*agree, and*R*is an equivalence relation on that set.^{[8]} - A right Euclidean relation is always quasitransitive,
^{[9]}as is a left Euclidean relation.^{[10]} - A connected right Euclidean relation is always transitive;
^{[11]}and so is a connected left Euclidean relation.^{[12]} - If
*X*has at least 3 elements, a connected right Euclidean relation*R*on*X*cannot be antisymmetric,^{[13]}and neither can a connected left Euclidean relation on*X*.^{[14]}On the 2-element set*X*= { 0, 1 }, e.g. the relation*xRy*defined by*y*=1 is connected, right Euclidean, and antisymmetric, and*xRy*defined by*x*=1 is connected, left Euclidean, and antisymmetric. - A relation
*R*on a set*X*is right Euclidean if, and only if, the restriction*R′*:=*R*|_{ran(R)}is an equivalence and for each*x*in*X*\ran(*R*), all elements to which*x*is related under*R*are equivalent under*R′*.^{[15]}Similarly,*R*on*X*is left Euclidean if, and only if,*R′*:=*R*|_{dom(R)}is an equivalence and for each*x*in*X*\dom(*R*), all elements that are related to*x*under*R*are equivalent under*R′*. - A left Euclidean relation is left-unique if, and only if, it is antisymmetric. Similarly, a right Euclidean relation is right unique if, and only if, it is anti-symmetric.
- A left Euclidean and left unique relation is vacuously transitive, and so is a right Euclidean and right unique relation.
- A left Euclidean relation is left quasi-reflexive. For left-unique relations, the converse also holds. Dually, each right Euclidean relation is right quasi-reflexive, and each right unique and right quasi-reflexive relation is right Euclidean.
^{[16]}