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

Let $a \in S$.

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

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

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