5/23/13 9:33 PM Partially ordered set - Wikipedia, the free encyclopediaPage 1 of 7https://en.wikipedia.org/wiki/Non-strict_order#Strict_and_non-strict_partial_orders The Hasse diagram of the set of all subsets of a three-element set {x, y, z}, ordered by inclusion.Partially ordered setFrom Wikipedia, the free encyclopedia(Redirected from Non-strict order)In mathematics, especially order theory, a partially ordered set (orposet ) formalizes and generalizes the intuitive concept of anordering, sequencing, or arrangement of the elements of a set. Aposet consists of a set together with a binary relation that indicatesthat, for certain pairs of elements in the set, one of the elementsprecedes the other. Such a relation is called a partial order to reflectthe fact that not every pair of elements need be related: for somepairs, it may be that neither element precedes the other in the poset.Thus, partial orders generalize the more familiar total orders, inwhich every pair is related. A finite poset can be visualized throughits Hasse diagram, which depicts the ordering relation.[1]A familiar real-life example of a partially ordered set is a collectionof people ordered by genealogical descendancy. Some pairs ofpeople bear the descendant-ancestor relationship, but other pairsbear no such relationship. Another real-life example is the set of all street addresses in a town: 50 Maple St. is clearly "before" 52 Maple St. and "after" 48 Maple St., therefore all Maple St. addresses are ordered. Also, all Pine St. addresses are ordered in the same way. But you cannot tell if 50 Maple St. is "before" or "after" 300Pine St. in this way, so any set that includes addresses on both streets is only partially ordered. The set of Maple St. addresses alone is a totally ordered set.Contents1 Formal definition2 Examples3 Extrema4 Orders on the Cartesian product of partially ordered sets5 Strict and non-strict partial orders6 Inverse and order dual7 Number of partial orders8 Linear extension9 In category theory10 Partial orders in topological spaces11 Interval12 See also13 Notes14 References15 External links
本文来源：https://www.2haoxitong.net/k/doc/707380d8c1c708a1284a4420.html
