Editing Completeness (order theory) (section)

===Least and greatest elements===

The easiest example of a supremum is the empty one, i.e. the supremum of the [[empty set]]. By definition, this is the least element among all elements that are greater than each member of the empty set. But this is just the [[least element]] of the whole poset, if it has one, since the empty subset of a poset ''P'' is conventionally considered to be both bounded from above and from below, with every element of ''P'' being both an upper and lower bound of the empty subset. Other common names for the least element are bottom and zero (0). The dual notion, the empty lower bound, is the [[greatest element]], top, or unit (1).

Posets that have a bottom are sometimes called pointed, while posets with a top are called unital or topped. An order that has both a least and a greatest element is bounded. However, this should not be confused with the notion of ''bounded completeness'' given below.