Definition:Upper Set
Definition
Let $\left({S, \preceq}\right)$ be an ordered set.
Let $U \subseteq S$.
Definition 1
$U$ is an upper set 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 set 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 set in $S$ if and only if:
- $U^\succeq = U$
where $U^\succeq$ is the upper closure of $U$.
Also known as
An upper set is also known as an upper-closed set or upward-closed set. Some sources call it an upset or up-set.
Also see
- Results about upper sets can be found here.
