A transitive relation is equivalently:1
- A non-unital $\mathbb {E}_{1}$-monoid in $\webleft (\mathrm{N}_{\bullet }\webleft (\mathbf{Rel}\webleft (A,A\webright )\webright ),\mathbin {\diamond }\webright )$;
- A non-unital monoid in $\webleft (\mathbf{Rel}\webleft (A,A\webright ),\mathbin {\diamond }\webright )$.
1Note that since $\mathbf{Rel}\webleft (A,A\webright )$ is posetal, transitivity is a property of a relation, rather than extra structure.
