By Miertus

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P/ has as many upper bounds in Q as p has upper bounds in P. The same statement holds for lower bounds. Proof. Let p 2 P and let x 2 P be such that x p. p/. Moreover, because ˚ is injective, no two upper bounds of p are mapped to the same point. p/ has at least as many upper bounds as p. p/ has an upper bound q that is not the image of an upper bound of p. p/, a contradiction. p/ are equal. To prove the same statement for lower bounds, we could simply follow the above argument with reversed comparabilities.

19. An ordered set P is called an antichain iff, for all p; q 2 P with p 6D q, we have p 6 q. 20. Let P be a finite ordered set and let A Â P be an antichain. Then there is an antichain B Â P that is maximal with respect to set inclusion and contains A. Antichains such as B are also called maximal antichains. Proof. Left to the reader as Exercise 2-13. Note that, for infinite ordered sets, this result is surprisingly challenging, requiring the use of Zorn’s Lemma (see Exercise 2-39). 5). 21. An ordered set C is a chain iff C does not contain any twoelement antichains.

If chains are totally ordered, one could say that antichains are totally unordered. 19. An ordered set P is called an antichain iff, for all p; q 2 P with p 6D q, we have p 6 q. 20. Let P be a finite ordered set and let A Â P be an antichain. Then there is an antichain B Â P that is maximal with respect to set inclusion and contains A. Antichains such as B are also called maximal antichains. Proof. Left to the reader as Exercise 2-13. Note that, for infinite ordered sets, this result is surprisingly challenging, requiring the use of Zorn’s Lemma (see Exercise 2-39).

### Combinatorial Chemistry and Technology by Miertus

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