Order Dimension - 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.

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  • Example (musician), a British musician
  • Example, Florida rock band For Squirrels' sole major-label album, released in 1995
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  • 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
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