An Entity of Type : owl:Thing, within Data Space : dbpedia.org associated with source document(s)

In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound. In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c. In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.

AttributesValues
rdfs:label
• Directed set
rdfs:comment
• In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound. In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c. In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.
sameAs
dct:subject
Wikipage page ID
Wikipage revision ID
Link from a Wikipage to another Wikipage
foaf:isPrimaryTopicOf
prov:wasDerivedFrom
has abstract
• In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound. In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c. The notion defined above is sometimes called an upward directed set. A downward directed set is defined analogously, meaning when every pair of elements is bounded below. Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Beware that other authors call a set directed if and only if it is directed both upward and downward. Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast partially ordered sets, which need not be directed). Join semilattices (which are partially ordered sets) are directed sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward. In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.
http://purl.org/voc/vrank#hasRank
http://purl.org/li...ics/gold/hypernym
is rdfs:seeAlso of
is Link from a Wikipage to another Wikipage of
Faceted Search & Find service v1.17_git21 as of Mar 09 2019

Alternative Linked Data Documents: PivotViewer | iSPARQL | ODE     Content Formats:       RDF       ODATA       Microdata      About

OpenLink Virtuoso version 07.20.3230 as of May 1 2019, on Linux (x86_64-generic-linux-glibc25), Single-Server Edition (61 GB total memory)