Category:Lower Sections

This category contains results about Lower Sections.
Definitions specific to this category can be found in Definitions/Lower Sections.

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 $L$.