Editing Completeness (order theory) (section)

==Relationships between completeness properties==

It was already observed that binary meets/joins yield all non-empty finite meets/joins. Likewise, many other (combinations) of the above conditions are equivalent.

* The best-known example is the existence of all suprema, which is in fact equivalent to the existence of all infima. Indeed, for any subset ''X'' of a poset, one can consider its set of lower bounds ''B''. The supremum of ''B'' is then equal to the infimum of ''X'': since each element of ''X'' is an upper bound of ''B'', sup&thinsp;''B'' is smaller than all elements of ''X'', i.e. sup&thinsp;''B'' is in ''B''. It is the greatest element of ''B'' and hence the infimum of ''X''. In a dual way, the existence of all infima implies the existence of all suprema.
* Bounded completeness can also be characterized differently. By an argument similar to the above, one finds that the supremum of a set with upper bounds is the infimum of the set of upper bounds. Consequently, bounded completeness is equivalent to the existence of all non-empty infima.
* A poset is a complete lattice [[if and only if]] it is a cpo and a join-semilattice. Indeed, for any subset ''X'', the set of all finite suprema (joins) of ''X'' is directed and the supremum of this set (which exists by directed completeness) is equal to the supremum of ''X''. Thus every set has a supremum and by the above observation we have a complete lattice. The other direction of the proof is trivial.
* Assuming the [[axiom of choice]], a poset is chain complete if and only if it is a dcpo.