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

## 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:

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*:

## Properties

- 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, also a non-symmetric relation can be both transitive and right Euclidean, for example,
*xRy*defined by*y*=0. - A relation which 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. - 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.^{[7]} - A right Euclidean relation is always quasitransitive,
^{[8]}and so is a left Euclidean relation.^{[9]} - A semi-connex right Euclidean relation is always transitive;
^{[10]}and so is a semi-connex left Euclidean relation.^{[11]} - If
*X*has at least 3 elements, a semi-connex right Euclidean relation*R*on*X*cannot be antisymmetric,^{[12]}and neither can a semi-connex left Euclidean relation on*X*.^{[13]}On the 2-element set*X*= { 0, 1 }, e.g. the relation*xRy*defined by*y*=1 is semi-connex, right Euclidean, and antisymmetric, and*xRy*defined by*x*=1 is semi-connex, 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’*.^{[14]}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.
^{[15]}

## References

- ^
^{a}^{b}Fagin, Ronald (2003),*Reasoning About Knowledge*, MIT Press, p. 60, ISBN 978-0-262-56200-3. **^**e.g. 0 ≤ 2 and 0 ≤ 1, but not 2 ≤ 1**^**e.g. 2*R*1 and 1*R*0, but not 2*R*0**^***xRy*and*xRx*implies*yRx*.**^**Equality of domain and range isn't necessary: the relation*xRy*defined by*y*=min{*x*,2} is right Euclidean on the natural numbers, and its range, {0,1,2}, is a proper subset of its domain, ℕ.**^**If*y*is in the range of*R*, then*xRy*∧*xRy*implies*yRy*, for some suitable*x*. This also proves that*y*is in the domain of*R*.**^**The*only if*direction follows from the previous paragraph. — For the*if*direction, assume*aRb*and*aRc*, then*a*,*b*,*c*are members of the domain and range of*R*, hence*bRc*by symmetry and transitivity; left Euclideanness of*R*follows similarly.**^**If*xRy*∧ ¬*yRx*∧*yRz*∧ ¬*zRy*holds, then both*y*and*z*are in the range of*R*. Since*R*is an equivalence on that set,*yRz*implies*zRy*. Hence the antecedent of the quasi-transitivity definion formula cannot be satisfied.**^**A similar argument applies, observing that*x*,*y*are in the domain of*R*.**^**If*xRy*∧*yRz*holds, then*y*and*z*are in the range of*R*. Since*R*is semi-connex,*xRz*or*zRx*or*x*=*z*holds. In case 1, nothing remains to be shown. In cases 2 and 3, also*x*is in the range. Hence,*xRz*follows from the symmetry and reflexivity of*R*on its range, respectively.**^**Similar, using that*x*,*y*are in the domain of*R*.**^**Since*R*is semi-connex, at least two distinct elements*x*,*y*are in its range, and*xRy*∨*yRx*holds. Since*R*is symmetric on its range, even*xRy*∧*yRx*holds. This contradicts the antisymmetry property.**^**By a similar argument, using the domain of*R*.**^***Only if:**R*’ is an equivalence as shown above. If*x*∈*X*\ran(*R*) and*xR’y*_{1}and*xR’y*_{2}, then*y*_{1}*Ry*_{2}by right Euclideaness, hence*y*_{1}*R’y*_{2}. —*If*: if*xRy*∧*xRz*holds, then*y*,*z*∈ran(*R*). In case also*x*∈ran(*R*), even*xR’y*∧*xR’z*holds, hence*yR’z*by symmetry and transitivity of*R’*, hence*yRz*. In case*x*∈*X*\ran(*R*), the elements*y*and*z*must be equivalent under*R’*by assumption, hence also*yRz*.**^**Jochen Burghardt (Nov 2018). Simple Laws about Nonprominent Properties of Binary Relations (Technical Report). arXiv:1806.05036v2. Lemma 44-46.