Definition:Lower Section/Definition 3

Definition

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

Let $L \subseteq S$.

$L$ is a lower section in $S$ if and only if:

$L^\preceq = L$

where $L^\preceq$ is the lower closure of $L$.

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.

Also see

• Equivalence of Definitions of Lower Section