\begin{definition}[Definition:Closure Operator/Ordering/Definition 1]
Let $\struct {S, \preceq}$ be an ordered set.
A '''closure operator''' on $S$ is a mapping:
:$\cl: S \to S$
which satisfies the following conditions for all elements $x, y \in S$:
{{begin-axiom}}
{{axiom | lc= $\cl$ is inflationary
        | ml= x
        | mo= \preceq 
        | mr= \map \cl x
}}
{{axiom | lc= $\cl$ is increasing
        | ml= x \preceq y
        | mo= \implies
        | mr= \map \cl x \preceq \map \cl y
}}
{{axiom | lc= $\cl$ is idempotent
        | ml= \map \cl {\map \cl x} 
        | mo= =
        | mr= \map \cl x
}}
{{end-axiom}}
\end{definition}