Definition:Ideal (Order Theory)/Proper Ideal

Definition

Let $\struct {S, \preccurlyeq}$ be an ordered set.

Let $\II$ be an ideal on $\struct {S, \preccurlyeq}$.

Then:

$\II$ is a proper ideal on $S$

if and only if:

$\II \ne S$

That is, if and only if $\II$ is a proper subset of $S$.

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