\begin{definition}[Definition:Preordering/Definition 1]
Let $\RR \subseteq S \times S$ be a relation on a set $S$.
$\RR$ is a '''preordering''' on $S$ {{iff}}:
{{begin-axiom}}
{{axiom | n = 1
        | lc= $\RR$ is reflexive
        | q = \forall a \in S
        | m = a \mathrel \RR a
}}
{{axiom | n = 2
        | lc= $\RR$ is transitive
        | q = \forall a, b, c \in S
        | m = a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c
}}
{{end-axiom}}
\end{definition}