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. 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.
Read more about Partially Ordered Set: Formal Definition, Examples, Extrema, Orders On The Cartesian Product of Partially Ordered Sets, Strict and Non-strict Partial Orders, Inverse and Order Dual, Number of Partial Orders, Linear Extension, In Category Theory, Partial Orders in Topological Spaces, Interval
Famous quotes containing the words partially, ordered and/or set:
“Some men have a necessity to be mean, as if they were exercising a faculty which they had to partially neglect since early childhood.”
—F. Scott Fitzgerald (1896–1940)
“The peace conference must not adjourn without the establishment of some ordered system of international government, backed by power enough to give authority to its decrees. ... Unless a league something like this results at our peace conference, we shall merely drop back into armed hostility and international anarchy. The war will have been fought in vain ...”
—Virginia Crocheron Gildersleeve (1877–1965)
“One does not set fire to a world which is already lost.”
—Friedrich Dürrenmatt (1921–1990)