Order Dimension - Example

Example

Let n be a positive integer, and let P be the partial order on the elements a_i and b_i (for 1 ≤ i ≤ n) in which a_i ≤ b_j whenever i ≠ j, but no other pairs are comparable. In particular, a_i and b_i 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 a_j except a_i (in some order), then includes b_i, then a_i, and ends with all the remaining b_j. 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 a_i preceding b_i, which would contradict the incomparability of a_i and b_i 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 S_n.