Category:Definitions/Lower Sections

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This category contains definitions related to Lower Sections.
Related results can be found in Category: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 $L$.