Partially Ordered Set

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 SetFormal 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

Other articles related to "partially ordered set, set, sets, partially, ordered":

Fence (mathematics)
... In mathematics, a fence, also called a zigzag poset, is a partially ordered set in which the order relations form a path with alternating orientations a < b > c < d > e < f > h ... A partially ordered set is series-parallel if and only if it does not have four elements forming a fence ... In this notation, a fence is a partially ordered set of the form Q(1,n) ...
Partially Ordered Set - Interval
... For a ≤ b, the closed interval is the set of elements x satisfying a ≤ x ≤ b (i.e ... a ≤ x and x ≤ b) ...
Pseudoideal - Basic Definitions
... LU(A) is the set of all lower bounds of the set of all upper bounds of the subset A of a partially ordered set ... A subset I of a partially ordered set (P,≤) is a Doyle pseudoideal, if the following condition holds A subset I of a partially ordered set (P,≤) is a pseudoideal, if the ...
Dedekind–Mac Neille Completion - Algorithms - Constructing The Set of Cuts
... Each cut of the augmented partial order, except for the one whose two sets intersect in the new element, is either a cut from the previous partial order or is formed by adding ... the problem of listing all cuts in a partially ordered set can be formulated as a special case of a simpler problem, of listing all maximal antichains in a different partially ... If P is any partially ordered set, let Q be a partial order whose elements contain two copies of P for each element x of P, Q contains two elements x0 and x1, with xi < yj if and only if x < y and i < j ...
Hausdorff Maximal Principle - Statement
... The Hausdorff maximal principle states that, in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset ... Here a maximal totally-ordered subset is one that, if enlarged in any way, does not remain totally ordered ... The maximal set produced by the principle is not unique, in general there may be many maximal totally ordered subsets containing a given totally ordered subset ...

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