Definition:Closure Operator/Ordering
< Definition:Closure Operator(Redirected from Definition:Closure Operator (Order Theory))
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Definition
Definition 1
Let $\struct {S, \preceq}$ be an ordered set.
A closure operator on $S$ is a mapping:
- $\cl: S \to S$
which satisfies the closure axioms as follows for all elements $x, y \in S$:
\((\text {cl} 1)\) | $:$ | $\cl$ is inflationary: | \(\ds x \) | \(\ds \preceq \) | \(\ds \map \cl x \) | ||||
\((\text {cl} 2)\) | $:$ | $\cl$ is increasing: | \(\ds x \preceq y \) | \(\ds \implies \) | \(\ds \map \cl x \preceq \map \cl y \) | ||||
\((\text {cl} 3)\) | $:$ | $\cl$ is idempotent: | \(\ds \map \cl {\map \cl x} \) | \(\ds = \) | \(\ds \map \cl x \) |
Definition 2
Let $\struct {S, \preceq}$ be an ordered set.
A closure operator on $S$ is a mapping:
- $\cl: S \to S$
which satisfies the following condition for all elements $x, y \in S$:
- $x \preceq \map \cl y \iff \map \cl x \preceq \map \cl y$
Also see
- Results about closure operators can be found here.