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?