Let $\preceq _{A}$ be a preorder on $A$, viewed also as an internal monad on $A$ via Proposition 8.5.4.1.1.
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1.
Left Modules. We have a natural identification
\[ \left\{ \text{Left modules over $\mathord {\preceq _{A}}$}\right\} \cong \left\{ \begin{aligned} & \text{Relations $R\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ such that,}\\ & \text{for each $b\in B$, the set $R(b)$ is}\\ & \text{upward-closed in $A$}\end{aligned} \right\} . \] -
2.
Right Modules. We have a natural identification
\[ \left\{ \text{Right modules over $\mathord {\preceq _{A}}$}\right\} \cong \left\{ \begin{aligned} & \text{Relations $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ such that,}\\ & \text{for each $b\in B$, the set $R^{-1}(b)$ is}\\ & \text{downward-closed in $A$}\end{aligned} \right\} . \] -
3.
Bimodules. We have a natural identification
\[ \left\{ \text{Bimodules over $\mathord {\preceq _{A}}$}\right\} \cong \left\{ \begin{gathered} \text{Quadruples $(B,C,R,S)$ such that:}\\ \begin{aligned} & \text{1. For each $b\in B$, the set $R(b)$ is}\\ & \phantom{\text{1. }}\text{upward-closed in $A$.}\\ & \text{2. For each $c\in C$, the set $S^{-1}(c)$ is}\\ & \phantom{\text{2. }}\text{downward-closed in $A$.}\end{aligned} \end{gathered} \right\} . \]
