Szpilrajn extension theorem

In order theory, the Szpilrajn extension theorem (also called the order-extension principle), proved by Edward Szpilrajn in 1930,[1] states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable. The theorem is one of many examples of the use of the axiom of choice in the form of Zorn's lemma to find a maximal set with certain properties.

Definitions and statement

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A binary relation   on a set   is formally defined as a set of ordered pairs   of elements of   and   is often abbreviated as  

A relation is reflexive if   holds for every element   it is transitive if   imply   for all   it is antisymmetric if   imply   for all   and it is a connex relation if   holds for all   A partial order is, by definition, a reflexive, transitive and antisymmetric relation. A total order is a partial order that is connex.

A relation   is contained in another relation   when all ordered pairs in   also appear in   that is,  implies   for all   The extension theorem states that every relation   that is reflexive, transitive and antisymmetric (that is, a partial order) is contained in another relation   which is reflexive, transitive, antisymmetric and connex (that is, a total order).

Proof

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The theorem is proved in two steps. First, one shows that, if a partial order does not compare some two elements, it can be extended to an order with a superset of comparable pairs. A maximal partial order cannot be extended, by definition, so it follows from this step that a maximal partial order must be a total order. In the second step, Zorn's lemma is applied to find a maximal partial order that extends any given partial order.

For the first step, suppose that a given partial order does not compare   and  . Then the order is extended by first adding the pair   to the relation, which may result in a non-transitive relation, and then restoring transitivity by adding all pairs   such that   This produces a relation that is still reflexive, antisymmetric and transitive and that strictly contains the original one. It follows that if the partial orders extending   are themselves partially ordered by extension, then any maximal element of this extension order must be a total order.

Next it is shown that the poset of partial orders extending  , ordered by extension, has a maximal element. The existence of such a maximal element is proved by applying Zorn's lemma to this poset. Zorn's lemma states that a partial order in which every chain has an upper bound has a maximal element. A chain in this poset is a set of relations in which, for every two relations, one extends the other. An upper bound for a chain   can be found as the union of the relations in the chain,  . This union is a relation that extends  , since every element of   is a partial order having   as a subset. Next, it is shown that   is a transitive relation. Suppose that   and   are in   so that there exist   such that   and  . Because   is a chain, one of   or   must extend the other and contain both   and  , and by its transitivity it also contains  , as does the union. Similarly, it can be shown that   is antisymmetric. Thus,   is an extension of  , so it belongs to the poset of extensions of  , and is an upper bound for  .

This argument shows that Zorn's lemma may be applied to the poset of extensions of  , producing a maximal element  . By the first step this maximal element must be a total order, completing the proof.

Strength

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Some form of the axiom of choice is necessary in proving the Szpilrajn extension theorem. The extension theorem implies the axiom of finite choice: if the union of a family of finite sets is given the empty partial order, and this is extended to a total order, the extension defines a choice from each finite set, its minimum element in the total order. Although finite choice is a weak version of the axiom of choice, it is independent of Zermelo–Fraenkel set theory without choice.[2]

The Szpilrajn extension theorem together with another consequence of the axiom of choice, the principle that every total order has a cofinal well-order, can be combined to prove the full axiom of choice. With these assumptions, one can choose an element from any given set by extending its empty partial order, finding a cofinal well-order, and choosing the minimum element from that well-ordering.[3]

Other extension theorems

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Arrow stated that every preorder (reflexive and transitive relation) can be extended to a total preorder (transitive and connex relation).[4] This claim was later proved by Hansson.[5][6]

Suzumura proved that a binary relation can be extended to a total preorder if and only if it is Suzumura-consistent, which means that there is no cycle of elements such that   for every pair of consecutive elements   and there is some pair of consecutive elements   in the cycle for which   does not hold.[6]

References

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  1. ^ Szpilrajn, Edward (1930), "Sur l'extension de l'ordre partiel" (PDF), Fundamenta Mathematicae (in French), 16: 386–389, doi:10.4064/fm-16-1-386-389
  2. ^ Moore, Gregory H. (1982), Zermelo's Axiom of Choice: Its Origins, Development, and Influence, Studies in the History of Mathematics and Physical Sciences, vol. 8, New York: Springer-Verlag, p. 222, doi:10.1007/978-1-4613-9478-5, ISBN 0-387-90670-3, MR 0679315
  3. ^ Howard, Paul; Rubin, Jean E. (1998), "Note 121", Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, vol. 59, Providence, Rhode Island: American Mathematical Society, p. 299, doi:10.1090/surv/059, ISBN 0-8218-0977-6, MR 1637107
  4. ^ Arrow, Kenneth J. (2012), "IV.3: Quasi-orderings and compatible weak orderings", Social Choice and Individual Values (3rd ed.), Yale University Press, p. 64, ISBN 978-0-300-18698-7
  5. ^ Hansson, Bengt (1968), "Choice structures and preference relations", Synthese, 18 (4): 443–458, doi:10.1007/BF00484979, JSTOR 20114617; see Lemma 3
  6. ^ a b Cato, Susumu (August 2011), "Szpilrajn, Arrow and Suzumura: concise proofs of extension theorems and an extension", Metroeconomica, 63 (2): 235–249, doi:10.1111/j.1467-999x.2011.04130.x