Category:Lower Closures

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


Let $\left({S, \preccurlyeq}\right)$ be an ordered set.

Let $a \in S$.


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

$a^\preccurlyeq := \left\{{b \in S: b \preccurlyeq a}\right\}$


That is, $a^\preccurlyeq$ is the set of all elements of $S$ that precede $a$.