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.
R
X
(x,y)
X,
(x,y)\inR
xRy.
A relation is reflexive if
xRx
x\inX;
xRyandyRz
xRz
x,y,z\inX;
xRyandyRx
x=y
x,y\inX;
xRyoryRx
x,y\inX.
A relation
R
S
R
S;
xRy
xSy
x,y\inX.
R
S
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
x
y
xRy
qRp
qRxandyRp.
R
Next it is shown that the poset of partial orders extending
R
l{C}
stylecupl{C}
R
l{C}
R
stylecupl{C}
(x,y)
(y,z)
stylecupl{C},
S,T\inl{C}
(x,y)\inS
(y,z)\inT
l{C}
S
T
(x,y)
(y,z)
(x,z)
stylecupl{C}
stylecupl{C}
R
R
l{C}
This argument shows that Zorn's lemma may be applied to the poset of extensions of
R
Q
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.
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
xRy
(x,y),
(x,y)
yRx