Definition:Strict Lower Closure/Element/Class Theory

Definition
Let $A$ be a class under an ordering $\preccurlyeq$.

Let $a \in A$.

The strict lower closure of $a$ (in $A$) is defined as:
 * $a^\prec := \set {b \in A: b \preccurlyeq a \land a \ne b}$

or:
 * $a^\prec := \set {b \in S: b \prec a}$

Also see

 * Definition:Lower Closure of Element (Class Theory)