nLab transitive relation

A transitive relation is a semicategory or magmoid $A$ such that for all objects $a$ and $b$ in $A$, there are no more than one morphism with domain $a$ and codomain $b$. Equivalently, a transitive relation is a semicategory or magmoid enriched on truth values.

Set theoretically, 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:

$$\forall (x, y, z: A),\; x \sim y \;\wedge\; y \sim z \;\Rightarrow\; x \sim z$$

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:

$$R^2 \subseteq R$$

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.