Definition:Upper Section/Definition 2

Definition

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

Let $U \subseteq S$.

$U$ is an upper section in $S$ if and only if:

$U^\succeq \subseteq U$

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

• Equivalence of Definitions of Upper Section