Category:Definitions/Upper Sections

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


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

Let $U \subseteq S$.


Definition 1

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

$\forall u \in U: \forall s \in S: u \preceq s \implies s \in U$


Definition 2

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

$U^\succeq \subseteq U$

where $U^\succeq$ is the upper closure of $U$.


Definition 3

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

$U^\succeq = U$

where $U^\succeq$ is the upper closure of $U$.