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( Partial order)
In mathematics, especially order theory, a partially ordered set (or poset) formalizes 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 describes, for certain pairs of elements in the set, the requirement that one of the elements must precede the other. However, a partially ordered set differs from a total order in that some pairs of elements may not be related to each other in this way. A finite poset can be visualized through its Hasse diagram, which depicts the ordering relation between certain pairs of elements and allows one to reconstruct the whole partial order structure. 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 ancestor-descendant relationship but, in general, other pairs bear no such relationship. A partial order is a binary relation "=" over a set P which is reflexive, antisymmetric, and transitive, i.e., for all a, b, and c in P, we have that In other words, a partial order is an antisymmetric preorder.
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Partial order Subcategories
Partial order Articles
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