Let $A$ be a set.
A transitive relation is equivalently:1
A non-unital $\mathbb {E}_{1}$-monoid in $(\mathrm{N}_{\bullet }(\mathbf{Rel}(A,A)),\mathbin {\diamond })$.
A non-unital monoid in $(\mathbf{Rel}(A,A),\mathbin {\diamond })$.
In detail, a relation $R$ on $A$ is transitive if we have an inclusion
of relations in $\mathbf{Rel}(A,A)$, i.e. if, for each $a,c\in A$, the following condition is satisfied:
Let $A$ be a set.
The set of transitive relations from $A$ to $B$ is the subset $\smash {\mathrm{Rel}^{\mathrm{trans}}(A)}$ of $\mathrm{Rel}(A,A)$ spanned by the transitive relations.
The poset of relations from $A$ to $B$ is is the subposet $\smash {\mathbf{Rel}^{\mathsf{trans}}(A)}$ of $\mathbf{Rel}(A,A)$ spanned by the transitive relations.
Let $R$ and $S$ be relations on $A$.
Interaction With Inverses. If $R$ is transitive, then so is $R^{\dagger }$.
Interaction With Composition. If $R$ and $S$ are transitive, then $S\mathbin {\diamond }R$ may fail to be transitive.
If $a\sim _{S\mathbin {\diamond }R}c$ and $c\sim _{S\mathbin {\diamond }r}e$, then:
There is some $b\in A$ such that:
$a\sim _{R}b$;
$b\sim _{S}c$;
There is some $d\in A$ such that:
$c\sim _{R}d$;
$d\sim _{S}e$.
