\begin{definition}[Definition:Ordering/Definition 1]
Let $S$ be a set.
An '''ordering on $S$''' is a relation $\RR$ on $S$ such that:
{{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
}}
{{axiom | n = 3
        | lc= $\RR$ is antisymmetric
        | q = \forall a \in S
        | m = a \mathrel \RR b \land b \mathrel \RR a \implies a = b
}}
{{end-axiom}}
\end{definition}