Partially ordered set
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
