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:
“There was an Old Man who supposed,
That the street door was partially closed;”
—Edward Lear (1812–1888)
“But one sound always rose above the clamor of busy life and, no matter how much of a tintinnabulation, was never confused and, for a moment lifted everything into an ordered sphere: that of the bells.”
—Johan Huizinga (1872–1945)
“1st Witch. When shall we three meet again?
In thunder, lightning, or in rain?
2nd Witch. When the hurly-burly’s done,
When the battle’s lost and won.
3rd Witch. That will be ere set of sun.
1st Witch. Where the place?
2nd Witch. Upon the heath.
3rd Witch. There to meet with Macbeth.”
—William Shakespeare (1564–1616)