Order Dimension - Example

Example

Let n be a positive integer, and let P be the partial order on the elements ai and bi (for 1 ≤ in) in which aibj whenever ij, but no other pairs are comparable. In particular, ai and bi are incomparable in P; P can be viewed as an oriented form of a crown graph. The illustration shows an ordering of this type for n = 4.

Then, for each i, any realizer must contain a linear order that begins with all the aj except ai (in some order), then includes bi, then ai, and ends with all the remaining bj. This is so because if there were a realizer that didn't include such an order, then the intersection of that realizer's orders would have ai preceding bi, which would contradict the incomparability of ai and bi in P. And conversely, any family of linear orders that includes one order of this type for each i has P as its intersection. Thus, P has dimension exactly n. In fact, P is known as the standard example of a poset of dimension n, and is usually denoted by Sn.

Read more about this topic:  Order Dimension

Other articles related to "example":

Example

Example may also refer to:

  • Example (musician), a British musician
  • Example, Florida rock band For Squirrels' sole major-label album, released in 1995
  • example.com, example.net, example.org, example.edu and .example, domain names reserved for use in documentation as examples
  • HMS Example (P165), an Archer-class patrol and training vessel of the British Royal Navy
  • The Example, a 1634 play by James Shirley
  • The Example (comics), a 2009 graphic novel by Tom Taylor and Colin Wilson
Finite Model Theory - Basic Challenges - Characterisation of A Class of Structures - Example
4 ... Next we have to scale the structures up by increasing m ...
Wave Front Set - Definition - Example
... Then the projection on the first component of a distribution's wave front set is nothing else than its classical singular support, i.e ... the complement of the set on which its restriction would be a smooth function ...

Famous quotes containing the word example:

    Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.
    Marcel Proust (1871–1922)