Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A (binary) relation $\sim$ on a set $A$ is transitive if in every chain of $3$ pairwise related elements, the first and last elements are also related:
which generalises from $3$ to any finite, positive number of elements.
In the language of the $2$-poset Rel of sets and relations, a relation $R: A \to A$ is transitive if it contains its composite with itself:
from which it follows that $R^n \subseteq R$ for any positive natural number $n$. To include the case where $n = 0$, we must explicitly state that the relation is reflexive.
Transitive relations are often understood as orders.