Definition:Lower Section/Definition 1
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Definition
Let $\struct {S, \preceq}$ be an ordered set.
Let $L \subseteq S$.
$L$ is a lower section in $S$ if and only if:
- $\forall l \in L, s \in S: s \preceq l \implies s \in L$
Also known as
A lower section is also known as a lower set.
Variants of this can also be seen: lower-closed set or downward-closed set.
Some sources call it a downset or down-set.
Sometimes the word section is understood, and such a collection referred to solely with the adjective lower.