**Example**

Let *n* be a positive integer, and let *P* be the partial order on the elements *a _{i}* and

*b*(for 1 ≤

_{i}*i*≤

*n*) in which

*a*≤

_{i}*b*whenever

_{j}*i*≠

*j*, but no other pairs are comparable. In particular,

*a*and

_{i}*b*are incomparable in

_{i}*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 *a _{j}* except

*a*(in some order), then includes

_{i}*b*, then

_{i}*a*, and ends with all the remaining

_{i}*b*. 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

_{j}*a*preceding

_{i}*b*, which would contradict the incomparability of

_{i}*a*and

_{i}*b*in

_{i}*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

*S*.

_{n}Read more about this topic: Order Dimension

### Other articles related to "example":

**Example**

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*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

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### 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)