Definition:Ordered Set of Closure Operators

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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.


Sources