Editing Completeness (order theory) (section)

{{Short description|Existence of certain infima or suprema of a given poset}}
In the [[mathematics|mathematical]] area of [[order theory]], '''completeness properties''' assert the existence of certain [[infimum|infima]] or [[supremum|suprema]] of a given [[partially ordered set]] (poset). The most familiar example is the [[completeness of the real numbers]]. A special use of the term refers to [[complete partial order]]s or [[complete lattice]]s. However, many other interesting notions of completeness exist.

The motivation for considering completeness properties derives from the great importance of [[suprema]] (least upper bounds, [[Join (mathematics)|joins]], "<math>\vee</math>") and [[infima]] (greatest lower bounds, [[meet (mathematics)|meets]], "<math>\wedge</math>") to the theory of partial orders. Finding a supremum means to single out one distinguished least element from the [[set (mathematics)|set]] of upper bounds. On the one hand, these special elements often embody certain concrete properties that are interesting for the given application (such as being the [[least common multiple]] of a set of numbers or the [[union (set theory)|union]] of a collection of sets). On the other hand, the knowledge that certain types of [[subset]]s are guaranteed to have suprema or infima enables us to consider the evaluation of these elements as ''total operations'' on a partially ordered set. For this reason, [[poset]]s with certain completeness properties can often be described as [[algebraic structure]]s of a certain kind. In addition, studying the properties of the newly obtained operations yields further interesting subjects.