Definition:Strict Lower Closure/Set

Definition
Let $\struct {S, \preceq}$ be an ordered set or a preordered set.

Let $T \subseteq S$.

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


 * $\ds T^\prec := \bigcup \set {t^\prec: t \in T}$

where $t^\prec$ denotes the strict lower closure of $t$ in $S$.

That is:
 * $T^\prec := \set {u \in S: \exists t \in T: u \prec t}$

Also see

 * Definition:Strict Lower Closure of Element


 * Definition:Strict Upper Closure of Subset
 * Definition:Lower Closure of Subset