Definition:Ordered Set of Closure Operators

Definition
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.