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:
“He who gives himself entirely to his fellow-men appears to them useless and selfish; but he who gives himself partially to them is pronounced a benefactor and philanthropist.”
—Henry David Thoreau (1817–1862)
“Twenty-four-hour room service generally refers to the length of time that it takes for the club sandwich to arrive. This is indeed disheartening, particularly when you’ve ordered scrambled eggs.”
—Fran Lebowitz (b. 1950)
“When you have come into the land that the LORD your God is giving you, and have taken possession of it and settled in it, and you say, “I will set a king over me, like all the nations that are around me,” you may indeed set over you a king whom the LORD your God will choose. One of your own community you may set as king over you; you are not permitted to put a foreigner over you, who is not of your own community.”
—Bible: Hebrew, Deuteronomy 17:14,15.