Completeness (order theory)

In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> area of <a href="/facts/Order_theory/gn6fM3oJ">order theory</a>, completeness properties assert the existence of certain <a href="/facts/Infimum/oXCKVopz">infima</a> or <a href="/facts/Supremum/oXCKVopz">suprema</a> of a given <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered set</a> (poset). The most familiar example is the <a href="/facts/Completeness_of_the_real_numbers/UQIn0qDw">completeness of the real numbers</a>. A special use of the term refers to <a href="/facts/Complete_partial_order/a855Q53A">complete partial orders</a> or <a href="/facts/Complete_lattice/pQbphWeR">complete lattices</a>. However, many other interesting notions of completeness exist.
The motivation for considering completeness properties derives from the great importance of <a href="/facts/Suprema/oXCKVopz">suprema</a> (least upper bounds, <a href="/facts/Join_(mathematics)/cYcLhRIR">joins</a>, "
 
 
 
 ∨
 
 
 {\displaystyle \vee }
 
") and <a href="/facts/Infima/oXCKVopz">infima</a> (greatest lower bounds, <a href="/facts/Meet_(mathematics)/cYcLhRIR">meets</a>, "
 
 
 
 ∧
 
 
 {\displaystyle \wedge }
 
") to the theory of partial orders. Finding a supremum means to single out one distinguished least element from the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> 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 <a href="/facts/Least_common_multiple/AX1jIUnw">least common multiple</a> of a set of numbers or the <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of a collection of sets). On the other hand, the knowledge that certain types of <a href="/facts/Subset/SJGERmbA">subsets</a> 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, <a href="/facts/Poset/qb4a3FAX">posets</a> with certain completeness properties can often be described as <a href="/facts/Algebraic_structure/TkuXca24">algebraic structures</a> of a certain kind. In addition, studying the properties of the newly obtained operations yields further interesting subjects.

Completeness (order theory) open-in-new

Completeness (order theory)