Category:Definitions/Lower Closures

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