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
- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices
- Mizar article WAYBEL10:def 1