# Partially Ordered Set

(Redirected from totally ordered set)

A Partially Ordered Set is a set whose set members are all in a partial order relation.

**AKA:**Poset.**Context:**- It can range from being a Finite Partially Ordered Set (finite poset) to being an Infinite Partially Ordered Set (infinite poset).
- It can range from being a Total Partially Ordered Set to being a Non-Total Partially Ordered Set.

**Example(s):**- An Acyclic Directed Graph, such as a hierarchy graph and a lattice graph.
- a Partial Order Tree.
- …

**Counter-Example(s):**- an Undirected Graph.
- a Total Order.

**See:**Preorder Relation, Order Theory, Relation (Mathematics), Hasse Diagram, Genealogy.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Glossary_of_order_theory#P Retrieved:2015-6-14.
- Partially ordered set. A partially ordered set (
*P*, ≤), or*poset*for short, is a set*P*together with a partial order ≤ on*P*. **Poset**. A partially ordered set.

- Partially ordered set. A partially ordered set (

### 2014

- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/partially_ordered_set Retrieved:2014-9-17.
- 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.

- In mathematics, especially order theory, a

### 2009

- http://en.wiktionary.org/wiki/partially_ordered_set
- 1. (set theory) A set having a specified partial order.
- 2. (set theory) Said set together with said partial order; the ordered pair of said set and said partial order.

- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Hierarchy_(mathematics)
- In mathematics, a hierarchy is a preorder, i.e. an ordered set. The term is used to stress a natural hierarchical relation among the elements. In particular, it is the preferred terminology for posets whose elements are classes of objects of increasing complexity. In that case, the preorder defining the hierarchy is the class-containment relation. Containment hierarchies are thus special cases of hierarchies.