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.