657,267
edits
No edit summary |
No edit summary |
||
Line 9: | Line 9: | ||
* <B>Counter-Example(s):</B> | * <B>Counter-Example(s):</B> | ||
** an [[Undirected Graph]]. | ** an [[Undirected Graph]]. | ||
* <B><U>See</U>:</B> [[Preorder Relation]]. | ** a [[Total Order]]. | ||
* <B><U>See</U>:</B> [[Preorder Relation]], [[Order Theory]], [[Relation (Mathematics)]], [[Hasse Diagram]], [[Genealogy]]. | |||
---- | ---- | ||
---- | ---- | ||
==References == | ==References == | ||
=== 2014 === | |||
* (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/partially_ordered_set Retrieved:2014-9-17. | |||
** In [[mathematics]], especially [[order theory]], a '''partially ordered set''' (or '''poset''') formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a [[Set (mathematics)|set]]. A poset consists of a set together with a [[Relation (mathematics)|binary relation]] that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. Such a relation is called a ''partial order'' to reflect the fact that not every pair of elements need be related: for some pairs, it may be that neither element precedes the other in the poset. <P> Thus, partial orders generalize the more familiar [[total order]]s, in which every pair is related. A finite poset can be visualized through its [[Hasse diagram]], which depicts the ordering relation. <P> A '''familiar''' real-life example of a partially ordered set is a collection of people ordered by [[genealogy|genealogical]] descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship. | |||
===2009=== | ===2009=== | ||
Line 19: | Line 25: | ||
*** 1. (set theory) A set having a specified partial order. | *** 1. (set theory) A set having a specified partial order. | ||
*** 2. (set theory) Said set together with said partial order; the ordered pair of said set and said partial order. | *** 2. (set theory) Said set together with said partial order; the ordered pair of said set and said partial order. | ||
<BR> | |||
* (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Hierarchy_(mathematics) | * (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Hierarchy_(mathematics) | ||
** In mathematics, a hierarchy is a preorder, i.e. an ordered set. The term is used to stress a natural hierarchical relation among the elements. In particular, it is the [[preferred term]]inology for posets whose elements are classes of objects of increasing complexity. In that case, the preorder defining the hierarchy is the class-containment relation. Containment hierarchies are thus special cases of hierarchies. | ** In mathematics, a hierarchy is a preorder, i.e. an ordered set. The term is used to stress a natural hierarchical relation among the elements. In particular, it is the [[preferred term]]inology for posets whose elements are classes of objects of increasing complexity. In that case, the preorder defining the hierarchy is the class-containment relation. Containment hierarchies are thus special cases of hierarchies. | ||
Line 29: | Line 33: | ||
__NOTOC__ | __NOTOC__ | ||
[[Category:Concept]] | [[Category:Concept]] | ||