\begin{definition}[Definition:Ordered Set of Closure Operators]
Let $L = \struct{S, \preceq}$ be an ordered set.
The '''ordered set of closure operators''' of $L$ is ordered subset of $\map {\operatorname{Increasing}} {L, L} = \struct{X, \preceq'}$
and is defined by
:$\map {\operatorname{Closure}} L := \struct{Y, \precsim}$
where
:$Y = \leftset{f:S \to S: f}$ is closure operator$\rightset{}$
:$\mathord\precsim = \mathord\preceq' \cap \paren{Y \times Y}$
:$\map {\operatorname{Increasing}} {L, L}$  denotes the ordered set of increasing mappings from $L$ into $L$.
$\map {\operatorname{Closure}} L$ as an ordered subset of an ordered set is an ordered set by Ordered Subset of Ordered Set is Ordered Set.
\end{definition}