Let n be a positive integer, and let P be the partial order on the elements ai and bi (for 1 ≤ i ≤ n) in which ai ≤ bj whenever i ≠ j, 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
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“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 (18711922)