Bounded complete poset

In the mathematical field of order theory, a partially ordered set is bounded complete if all of its subsets that have some upper bound also have a least upper bound. Such a partial order can also be called consistently or coherently complete (Visser 2004, p. 182), since any upper bound of a set can be interpreted as some consistent (non-contradictory) piece of information that extends all the information present in the set. Hence the presence of some upper bound in a way guarantees the consistency of a set. Bounded completeness then yields the existence of a least upper bound of any "consistent" subset, which can be regarded as the most general piece of information that captures all the knowledge present within this subset. This view closely relates to the idea of information ordering that one typically finds in domain theory.

Formally, a partially ordered set (P, ≤) is bounded complete if the following holds for any subset S of P:

Bounded completeness has various relationships to other completeness properties, which are detailed in the article on completeness in order theory. The term bounded poset is sometimes used to refer to a partially ordered set that has both a least and a greatest element. Hence it is important to distinguish between a bounded-complete poset and a bounded complete partial order (cpo).