left and right euclidean;
A (binary) relation on a set is transitive if in every chain of pairwise related elements, the first and last elements are also related:
\forall (x, y, z: A),\; x \sim y \;\wedge\; y \sim z \;\Rightarrow\; x \sim z
which generalises from to any finite, positive number of elements.
In the language of the -poset Rel of sets and relations, a relation is transitive if it contains its composite with itself:
R^2 \subseteq R
Transitive relations are often understood as orders.