Lower Section with no Maximal Element
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Theorem
Let $\struct {S, \preceq}$ be an ordered set.
Let $L \subseteq S$.
Then:
- $L$ is a lower section in $S$ with no maximal element
- $\ds L = \bigcup \set {l^\prec: l \in L}$
where $l^\prec$ is the strict lower closure of $l$.
Proof
- Lower section is dual to upper section.
- Maximal element is dual to minimal element.
- Strict lower closure is dual to strict upper closure.
Thus the theorem holds by the duality principle applied to Upper Section with no Minimal Element.
$\blacksquare$