About: Completeness (order theory)

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In the mathematical area of order theory, completeness properties assert the existence of certain infima or 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 orders or complete lattices. However, many other interesting notions of completeness exist.

Attributes	Values
rdfs:label	Ordnungsvollständigkeit (de) Completeness (order theory) (en)
rdfs:comment	Ordnungsvollständigkeit ist ein Begriff aus der Algebra, speziell der Körpertheorie, der aber für beliebige geordnete Mengen definiert werden kann. Der Begriff der Ordnungsvollständigkeit erweist sich in der Ordnungstopologie für nicht zu „große“ geordnete Mengen als verwandt mit dem Begriff der Vollständigkeit in metrischen Räumen. (de) In the mathematical area of order theory, completeness properties assert the existence of certain infima or 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 orders or complete lattices. However, many other interesting notions of completeness exist. (en)
dcterms:subject	Order theory
Wikipage page ID	549445 (xsd:integer)
Wikipage revision ID	1069867853 (xsd:integer)
Link from a Wikipage to another Wikipage	Product order Algebraic structure Join (mathematics) Universal algebra Limit-preserving function (order theory) Order theory Completeness of the real numbers Mathematical induction Mathematics Meet (mathematics) Order theory Class (set theory) Function (mathematics) Galois connection Morphism Complete lattice Complete partial order Completely distributive lattice Empty set Total order Distributivity (order theory) Domain theory Duality (order theory) Lattice (order) Greatest element Finite set Partially ordered set Directed set Directed complete partial order Heyting algebra Least common multiple Suprema Supremum Axiom of choice Boolean algebra (structure) Poset Ideal (order theory) Identity (mathematics) If and only if Category theory Semilattice Set (mathematics) Chain complete Infimum Union (set theory) Completion (order theory) Subset Lower set Powerset Least element Infima Bounded complete dbr:Categorical_formulation_of_order_theory
sameAs	Completeness (order theory) Completeness (order theory) Completeness (order theory) Completeness (order theory)
dbp:wikiPageUsesTemplate	dbt:Nobreak
has abstract	Ordnungsvollständigkeit ist ein Begriff aus der Algebra, speziell der Körpertheorie, der aber für beliebige geordnete Mengen definiert werden kann. Der Begriff der Ordnungsvollständigkeit erweist sich in der Ordnungstopologie für nicht zu „große“ geordnete Mengen als verwandt mit dem Begriff der Vollständigkeit in metrischen Räumen. (de) In the mathematical area of order theory, completeness properties assert the existence of certain infima or 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 orders or complete lattices. However, many other interesting notions of completeness exist. The motivation for considering completeness properties derives from the great importance of suprema (least upper bounds, joins, "") and infima (greatest lower bounds, meets, "") to the theory of partial orders. Finding a supremum means to single out one distinguished least element from the 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 of a collection of sets). On the other hand, the knowledge that certain types of subsets are guaranteed to have suprema or infima enables us to consider the computation of these elements as total operations on a partially ordered set. For this reason, posets with certain completeness properties can often be described as algebraic structures of a certain kind. In addition, studying the properties of the newly obtained operations yields further interesting subjects. (en)
prov:wasDerivedFrom	wikipedia-en:Completeness_(order_theory)?oldid=1069867853&ns=0
page length (characters) of wiki page	13167 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Completeness_(order_theory)
is Link from a Wikipage to another Wikipage of	Binary relation Dedekind cut Infimum and supremum Interior algebra Real analysis Lexicographic order topology on the unit square Limit-preserving function (order theory) List of order theory topics List of real analysis topics Completeness of the real numbers Connected relation Order theory Galois connection Bounded complete poset Sigma-ideal Compact element Complete Heyting algebra Complete lattice Complete partial order Denotational semantics of the Actor model Total order Distributivity (order theory) Domain theory Join and meet Lattice (order) Linear continuum Cyclic order Glossary of order theory Rational choice theory Heyting algebra

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