Let $A$ be a set.
We have a natural identification1
\[ \webleft\{ \begin{gathered} \text{Monads in}\\ \text{$\textbf{Rel}$ on $A$} \end{gathered} \webright\} \cong \webleft\{ \text{Preorders on $A$}\webright\} . \]
A monad in $\textbf{Rel}$ on $A$ consists of a relation $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ together with maps
\begin{align*} \mu _{R} & \colon R\mathbin {\diamond }R \subset R,\\ \eta _{R} & \colon \chi _{A} \subset R \end{align*}
making the diagrams
commute. However, since all morphisms involved are inclusions, the commutativity of the above diagrams is automatic, and hence all that is left is the data of the two maps $\mu _{R}$ and $\eta _{R}$, which correspond respectively to the following conditions: - For each $a,b,c\in A$, if $a\sim _{R}b$ and $b\sim _{R}c$, then $a\sim _{R}c$.
- For each $a\in A$, we have $a\sim _{R}a$.
These are exactly the requirements for $R$ to be a preorder (
). Conversely, any preorder $\preceq $ gives rise to a pair of maps $\mu _{\preceq }$ and $\eta _{\preceq }$, forming a monad on $A$.
