nLab euclidean relation

Euclidean relations

Euclidean relations


A (binary) relation \sim on an abstract set AA is left euclidean if every two elements both related to a third are also related to each other:

x,y,z:A,xz,yzxy. x, y, z: A,\; x \sim z,\; y \sim z \;\vdash\; x \sim y .

A relation \sim is right euclidean if this works in the other order:

x,y,z:A,xy,xzyz. x, y, z: A,\; x \sim y,\; x \sim z \;\vdash\; y \sim z .

One could also say euclidean and co-euclidean or op-euclidean, or similarly.

Relationship to other kinds of relations

An equivalence relation is a relation that is both reflexive and euclidean in either direction (in which case it's euclidean in both directions). Defining equivalence relations this way (rather than using transitivity and symmetry) is analogous to defining a group using (one-sided) division instead of multiplication and inverse; both are special cases of an analogous definition of groupoid. One can also define an equivalence relation as a relation that is both entire and left euclidean (or the reverse); this is analogous to defining a group using division and furthermore merely stating that some element exists rather than that there is an identity element. (Perhaps more familiarly, one can define a subgroup of a given group as a subset that is closed under division and has at least one element.)

A circular relation? satisfies

x,y,z:A,xy,yzzx. x, y, z: A,\; x \sim y,\; y \sim z \;\vdash\; z \sim x .

This is analogous to taking the reciprocal of the product in a group(oid). Again, a reflexive circular relation is an equivalence relation.


The above (reflexive and euclidean) is the earliest definition of equivalence, dating back (at least) to Euclid's Common Notion 1 (hence the name). In Heath's translation,

Things [geometric figures] which are equal to [have the same magnitude as] the same thing are also equal to one another.

It is not clear, however, if Euclid appreciated the distinction between this and transitivity; he may have taken it for granted that equality of magnitude is symmetric, which destroys the distinction. (One can similarly argue whether Common Notion 4 expresses reflexivity or whether it merely means that congruence entails equality of magnitude, with the equivalence-relation nature of congruence again going unstated. See this discussion.)

Last revised on July 15, 2023 at 00:06:25. See the history of this page for a list of all contributions to it.