Editing Completeness (order theory) (section)

===Further completeness conditions===

The strongest form of completeness is the existence of all suprema and all infima. The posets with this property are the [[complete lattice]]s. However, using the given order, one can restrict to further classes of (possibly infinite) subsets, that do not yield this strong completeness at once.

If all [[directed set|directed subsets]] of a poset have a supremum, then the order is a [[directed complete partial order|directed-complete partial order]] (dcpo). These are especially important in [[domain theory]]. The seldom-considered dual notion to a dcpo is the filtered-complete poset. Dcpos with a least element ("pointed dcpos") are one of the possible meanings of the phrase [[complete partial order]] (cpo).

If every subset that has ''some'' upper bound has also a least upper bound, then the respective poset is called [[bounded complete]]. The term is used widely with this definition that focuses on suprema and there is no common name for the dual property. However, bounded completeness can be expressed in terms of other completeness conditions that are easily dualized (see below). Although concepts with the names "complete" and "bounded" were already defined, confusion is unlikely to occur since one would rarely speak of a "bounded complete poset" when meaning a "bounded cpo" (which is just a "cpo with greatest element"). Likewise, "bounded complete lattice" is almost unambiguous, since one would not state the boundedness property for complete lattices, where it is implied anyway. Also note that the empty set usually has upper bounds (if the poset is non-empty) and thus a bounded-complete poset has a least element.

One may also consider the subsets of a poset which are [[total order|totally ordered]], i.e. the [[Total order#Chains|chains]]. If all chains have a supremum, the order is called [[chain complete]]. Again, this concept is rarely needed in the dual form.