Szpilrajn extension theorem
Introduction In order theory, the '''Szpilrajn extension theorem''' (also called the '''order-extension principle'''), proved by Edward Szpilrajn in 1930, 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 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 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 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.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.
Other extension theorems Arrow stated that every preorder (reflexive and transitive relation) can be extended to a total preorder (transitive and connex relation). This claim was later proved by Hansson.Suzumura proved that a binary relation can be extended to a total preorder if and only if it is , 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.
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