# nLab euclidean relation

## In higher category theory

A (binary) relation $\sim$ on a set $A$ is left euclidean if in any two elements both related to a third are also related to each other:

$\forall \left(x,y,z:A\right),\phantom{\rule{thickmathspace}{0ex}}x\sim z\phantom{\rule{thickmathspace}{0ex}}\wedge \phantom{\rule{thickmathspace}{0ex}}y\sim z\phantom{\rule{thickmathspace}{0ex}}⇒\phantom{\rule{thickmathspace}{0ex}}x\sim y$\forall (x, y, z: A),\; x \sim z \;\wedge\; y \sim z \;\Rightarrow\; x \sim y

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

$\forall \left(x,y,z:A\right),\phantom{\rule{thickmathspace}{0ex}}x\sim y\phantom{\rule{thickmathspace}{0ex}}\wedge \phantom{\rule{thickmathspace}{0ex}}x\sim z\phantom{\rule{thickmathspace}{0ex}}⇒\phantom{\rule{thickmathspace}{0ex}}y\sim z$\forall (x, y, z: A),\; x \sim y \;\wedge\; x \sim z \;\Rightarrow\; y \sim z

## Examples

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; 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.)

## History

This is the earliest notion of equivalence, dating back (at least) to Euclid's Common Notion 1. 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.)

Revised on August 24, 2012 20:05:00 by Urs Schreiber (89.204.138.8)