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.

Formal definition

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Partially_ordered_set

List of order structures in mathematics

In mathematics, and more particularly in order theory, several different types of ordered set have been studied. They include:

Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise

Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list.

Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be

Preorders, a generalization of partial orders allowing ties (represented as equivalences and distinct from incomparabilities)

Semiorders, partial orders determined by comparison of numerical values, in which values that are too close to each other are incomparable; a subfamily of partial orders with certain restrictions

Total orders, orderings that specify, for every two distinct elements, which one is less than the other

Weak orders, generalizations of total orders allowing ties (represented either as equivalences or, in strict weak orders, as transitive incomparabilities)

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/List_of_order_structures_in_mathematics