Definition:Lower Closure/Element/Also known as
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Lower Closure of Element: Also known as
The lower closure of an element $a$ is also known as:
- the down-set of $a$
- the down set of $a$
- the lower set of $a$
- the lower section of $a$
- the set of preceding elements to $a$
The terms weak lower closure and weak down-set are also encountered, so as explicitly to distinguish this from the strict lower closure of $a$.
When $\preccurlyeq$ is a well-ordering, the term weak initial segment is often used, and defined as a separate concept in its own right.
The notations $S_a$ or $\bar S_a$ are frequently then seen.
Some authors use the term (weak) initial segment to refer to the lower closure with respect to a general ordering.