Difference between revisions of "Partially ordered set"

Latest revision as of 09:25, 17 September 2016

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 elements of a 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

• Partially ordered set @ Wikipedia.org