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- Q554403 subject Q7451460.
- Q554403 subject Q7452041.
- Q554403 abstract "In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for P upside down, will indeed yield a partially ordered set. In a broader sense, two posets are also said to be duals if they are dually isomorphic, i.e. if one poset is order isomorphic to the dual of the other.The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets: If a given statement is valid for all partially ordered sets, then its dual statement, obtained by inverting the direction of all order relations and by dualizing all order theoretic definitions involved, is also valid for all partially ordered sets.If a statement or definition is equivalent to its dual then it is said to be self-dual. Note that the consideration of dual orders is so fundamental that it often occurs implicitly when writing ≥ for the dual order of ≤ without giving any prior definition of this "new" symbol.".
- Q554403 thumbnail Duale_Verbaende.svg?width=300.
- Q554403 wikiPageWikiLink Q1052692.
- Q554403 wikiPageWikiLink Q10564851.
- Q554403 wikiPageWikiLink Q1069998.
- Q554403 wikiPageWikiLink Q1196892.
- Q554403 wikiPageWikiLink Q1248241.
- Q554403 wikiPageWikiLink Q130998.
- Q554403 wikiPageWikiLink Q1475294.
- Q554403 wikiPageWikiLink Q1656686.
- Q554403 wikiPageWikiLink Q17502105.
- Q554403 wikiPageWikiLink Q194404.
- Q554403 wikiPageWikiLink Q2362924.
- Q554403 wikiPageWikiLink Q2363730.
- Q554403 wikiPageWikiLink Q2528262.
- Q554403 wikiPageWikiLink Q3491411.
- Q554403 wikiPageWikiLink Q395.
- Q554403 wikiPageWikiLink Q474715.
- Q554403 wikiPageWikiLink Q4973304.
- Q554403 wikiPageWikiLink Q583760.
- Q554403 wikiPageWikiLink Q595364.
- Q554403 wikiPageWikiLink Q6564405.
- Q554403 wikiPageWikiLink Q7451460.
- Q554403 wikiPageWikiLink Q7452041.
- Q554403 wikiPageWikiLink Q901446.
- Q554403 wikiPageWikiLink Q912887.
- Q554403 wikiPageWikiLink Q949972.
- Q554403 wikiPageWikiLink Q997521.
- Q554403 comment "In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for P upside down, will indeed yield a partially ordered set.".
- Q554403 label "Duality (order theory)".
- Q554403 depiction Duale_Verbaende.svg.