Definition:Ordered Set of Closure Operators

Definition
Let $L = \left({S, \preceq}\right)$ be an ordered set.

The ordered set of closure operators of $L$ is ordered subset of $\operatorname{Increasing} \left({L, L}\right) = \left({X, \preceq'}\right)$ and is defined by
 * $\operatorname{Closure}\left({L}\right) := \left({Y, \precsim}\right)$

where
 * $Y = \left\{ {f:S \to S: f}\right.$ is closure operator$\left.\right\}$
 * $\mathord\precsim = \mathord\preceq' \cap \left({Y \times Y}\right)$
 * $\operatorname{Increasing} \left({L, L}\right)$ denotes the ordered set of increasing mappings from $L$ into $L$.

$\operatorname{Closure}\left({L}\right)$ as an ordered subset of an ordered set is an ordered set by Ordered Subset of Ordered Set is Ordered Set.