| dbo:abstract
|
- In order theory, a discipline within mathematics, a critical pair is a pair of elements in a partially ordered set that are incomparable but that could be made comparable without requiring any other changes to the partial order. Formally, let P = (S, ≤) be a partially ordered set. Then a critical pair is an ordered pair (x, y) of elements of S with the following three properties:
* x and y are incomparable in P,
* for every z in S, if z < x then z < y, and
* for every z in S, if y < z then x < z. If (x, y) is a critical pair, then the binary relation obtained from P by adding the single relationship x ≤ y is also a partial order. The properties required of critical pairs ensure that, when the relationship x ≤ y is added, the addition does not cause any violations of the transitive property. A set R of linear extensions of P is said to reverse a critical pair (x, y) in P if there exists a linear extension in R for which y occurs earlier than x. This property may be used to characterize realizers of finite partial orders: A nonempty set R of linear extensions is a realizer if and only if it reverses every critical pair. (en)
|
| dbo:thumbnail
| |
| dbo:wikiPageID
| |
| dbo:wikiPageLength
|
- 2098 (xsd:nonNegativeInteger)
|
| dbo:wikiPageRevisionID
| |
| dbo:wikiPageWikiLink
| |
| dbp:wikiPageUsesTemplate
| |
| dcterms:subject
| |
| gold:hypernym
| |
| rdf:type
| |
| rdfs:comment
|
- In order theory, a discipline within mathematics, a critical pair is a pair of elements in a partially ordered set that are incomparable but that could be made comparable without requiring any other changes to the partial order. Formally, let P = (S, ≤) be a partially ordered set. Then a critical pair is an ordered pair (x, y) of elements of S with the following three properties:
* x and y are incomparable in P,
* for every z in S, if z < x then z < y, and
* for every z in S, if y < z then x < z. (en)
|
| rdfs:label
|
- Critical pair (order theory) (en)
|
| owl:sameAs
| |
| prov:wasDerivedFrom
| |
| foaf:depiction
| |
| foaf:isPrimaryTopicOf
| |
| is dbo:wikiPageDisambiguates
of | |
| is dbo:wikiPageWikiLink
of | |
| is foaf:primaryTopic
of | |
.svg?width=300)
.svg){kind=link}
.svg){kind=link}