Karl Jones: Created page with "In mathematics, especially in order theory, a '''partially ordered set''' (or '''poset''') formalizes and generalizes the intuitive concept of an ordering, sequencing,..."

2016-09-17T16:25:41Z

Created page with "In mathematics, especially in order theory, a '''partially ordered set''' (or '''poset''') formalizes and generalizes the intuitive concept of an ordering, sequencing,..."

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In [[mathematics]], especially in [[order theory]], a '''partially ordered set''' (or '''poset''') formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the [[Element (mathematics)|elements]] of a [[Set (mathematics)|set]].

== Description ==

A poset consists of a set together with a [[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. Thus, partial orders generalize the more familiar total orders, in which every pair is related.

A finite poset can be visualized through its [[Hasse diagram]], which depicts the ordering relation.

A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship.

== See also ==

* [[Antimatroid]], a formalization of orderings on a set that allows more general families of orderings than posets
* [[Causal set]]
* [[Comparability graph]]
* [[Complete partial order]]
* [[Directed set]]
* [[Element (mathematics)]]
* [[Graded poset]]
* [[Hasse diagram]]
* [[Incidence algebra]]
* [[Lattice]]
* [[Locally finite poset]]
* [[Möbius function on posets]]
* [[Ordered group]]
* [[Poset topology]] - a kind of topological space that can be defined from any poset
* [[Scott continuity]] - continuity of a function between two partial orders.
* [[Semilattice]]
* [[Semiorder]]
* [[Stochastic dominance]]
* [[Strict weak ordering]] – strict partial order "<" in which the relation "neither a < b nor b < a" is transitive.
* [[Total order]]
* [[Zorn's lemma]]

== External links ==

* [https://en.wikipedia.org/wiki/Partially_ordered_set Partially ordered set] @ Wikipedia.org

[[Category:Mathematics]]
[[Category:Set theory]]

Partially ordered set - Revision history

Karl Jones: Created page with "In mathematics, especially in order theory, a '''partially ordered set''' (or '''poset''') formalizes and generalizes the intuitive concept of an ordering, sequencing,..."