Category:Upper Closures

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This category contains results about Upper Closures in the context of Order Theory.
Definitions specific to this category can be found in Definitions/Upper Closures.

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

Let $a \in S$.

The upper closure of $a$ (in $S$) is defined as:

$a^\succcurlyeq := \set {b \in S: a \preccurlyeq b}$

That is, $a^\succcurlyeq$ is the set of all elements of $S$ that succeed $a$.