Definition:Upper Section
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This page is about Upper Section. For other uses, see Section.
Definition
Let $\struct {S, \preceq}$ be an ordered set.
Let $U \subseteq S$.
Definition 1
$U$ is an upper section in $S$ if and only if:
- $\forall u \in U: \forall s \in S: u \preceq s \implies s \in U$
Definition 2
$U$ is an upper section in $S$ if and only if:
- $U^\succeq \subseteq U$
where $U^\succeq$ is the upper closure of $U$.
Definition 3
$U$ is an upper section in $S$ if and only if:
- $U^\succeq = U$
where $U^\succeq$ is the upper closure of $U$.
Also known as
An upper section is also known as an upper set.
Variants of this can also be seen: upper-closed set or upward-closed set.
Some sources call it an upset or up-set.
Sometimes the word section is understood, and such a collection referred to solely with the adjective upper.
Also see
- Results about upper sections can be found here.