Definition:Lower Section

This page is about Lower Section. For other uses, see Section.

Definition

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

Let $L \subseteq S$.

Definition 1

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

$\forall l \in L, s \in S: s \preceq l \implies s \in L$

Definition 2

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

$L^\preceq \subseteq L$

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

Definition 3

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

$L^\preceq = L$

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

Class Theory

In the context of class theory, the definition follows the same lines:

Let $A$ be a class under a total ordering $\preccurlyeq$.

Let $L$ be a subclass of $A$ such that:

$\forall x \in L: \forall a \in A \setminus L: x \preccurlyeq a$

where $A \setminus L$ is the difference between $A$ and $L$.

Then $L$ is known as a lower section of $A$.

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

• Definition:Lower Closure
• Definition:Upper Section

• Results about lower sections can be found here.