# Non-transitive euclidean relation

A relation $R$ on a set $X$ is called Euclidean if for all $a, b, c \in X$, if $aRb$ and $aRc$ then $bRc$.

This does not imply transitivity - the relation $\{(a, b), (b, c), (b, b), (c, c), (c, b)\}$ is non-transitive and Euclidean.

I was wondering if there are any "real life examples" of these kinds of relations, which are interesting too?

I don't know if it counts as an interesting real-life example, but the relation on $\{0,1,-1\}$ defined by $xRy$ if $xy\leq0$ is at least an example. Further, it seems that this generalises to any set made up of an arbitrary amount of positive numbers, $0$, and precisely one negative number. It might generalise to other sets, but I'm pretty confident about that one.
Edit: I retract my statement, since $0R1$ and $0R1$ but $1\not R1$
$x\text{R}y$ defined by $0≤x≤y+1≤2$ is not transitive, but right Euclidean on natural numbers.