In

order theory
Order theory is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...

, the Szpilrajn extension theorem (also called the order-extension principle), proved by Edward Szpilrajn in 1930,. states that every strict partial order
In mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is contained in a total order
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. 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 mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

in the form of Zorn's lemma
Zorn's lemma, also known as the Kuratowskiâ€“Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...

to find a maximal set with certain properties.
Definitions and statement

Abinary relation
Binary may refer to:
Science and technology
Mathematics
* Binary number
In mathematics and digital electronics
Digital electronics is a field of electronics
The field of electronics is a branch of physics and electrical engineeri ...

$R$ on a set $X$ is formally defined as a set of ordered pairs $(x,\; y)$ of elements of $X,$ and $(x,\; y)\; \backslash in\; R$ is often abbreviated as $xRy.$
A relation is reflexive if $xRx$ holds for every element $x\; \backslash in\; X;$ it is transitive
Transitivity or transitive may refer to:
Grammar
* Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects
* Transitive verb, a verb which takes an object
* Transitive case, a grammatical case to mark arg ...

if $xRy\; \backslash text\; yRz$ imply $xRz$ for all $x,\; y,\; z\; \backslash in\; X;$ it is antisymmetric if $xRy\; \backslash text\; yRx$ imply $x\; =\; y$ for all $x,\; y\; \backslash in\; X;$ and it is a connex relation
In mathematics, a homogeneous relation is called a connex relation, or a relation having the property of connexity, if it relates all pairs of elements in some way. More formally, the homogeneous relation on a set is connex when for all x, y \in ...

if $xRy\; \backslash text\; yRx$ holds for all $x,\; y\; \backslash in\; X.$ A partial order is, by definition, a reflexive, transitive and antisymmetric relation. A total order is a partial order that is connex.
A relation $R$ is contained in another relation $S$ when all ordered pairs in $R$ also appear in $S;$ that is,$xRy$ implies $xSy$ for all $x,\; y\; \backslash in\; X.$ The extension theorem states that every relation $R$ that is reflexive, transitive and antisymmetric (that is, a partial order) is contained in another relation $S$ which is reflexive, transitive, antisymmetric and connex (that is, a total order).
Proof

The theorem is proved in two steps. First, if a partial order does not compare $x$ and $y,$ it can be extended by first adding the pair $(x,\; y)$ and then performing thetransitive closure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, and second, since this operation generates an ordering that strictly contains the original one and can be applied to all pairs of incomparable elements, there exists a relation in which all pairs of elements have been made comparable.
The first step is proved as a preliminary lemma, in which a partial order where a pair of elements $x$ and $y$ are incomparable is changed to make them comparable. This is done by first adding the pair $xRy$ to the relation, which may result in a non-transitive relation, and then restoring transitivity by adding all pairs $qRp$ such that $qRx\; \backslash text\; yRp.$ This is done on a single pair of incomparable elements $x$ and $y,$ and produces a relation that is still reflexive, antisymmetric and transitive and that strictly contains the original one.
Next it is shown that the poset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of partial orders containing $R,$ ordered by inclusion, has a maximal element. The existence of such a maximal element is proved by applying Zorn's lemma
Zorn's lemma, also known as the Kuratowskiâ€“Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...

to this poset. A chain in this poset is a set of relations containing $R$ such that given any two of these relations, one is contained in the other.
To apply Zorn's lemma, it must be shown that every chain has an upper bound in the poset. Let $\backslash mathcal$ be such a chain, and it remains to show that the union of its elements, $\backslash bigcup\; \backslash mathcal,$ is an upper bound for $\backslash mathcal$ which is in the poset: $\backslash mathcal$ contains the original relation $R$ since every element of $\backslash mathcal$ is a partial order containing $R.$ Next, it is shown that $\backslash bigcup\; \backslash mathcal$ is a transitive relation. Suppose that $(x,\; y)$ and $(y,\; z)$ are in $\backslash bigcup\; \backslash mathcal,$ so that there exist $S,\; T\; \backslash in\; \backslash mathcal$ such that $(x,\; y)\; \backslash in\; S\; \backslash text\; (y,\; z)\; \backslash in\; T.$ Since $\backslash mathcal$ is a chain, either $S\; \backslash subseteq\; T\; \backslash text\; T\; \backslash subseteq\; S.$ Suppose $S\; \backslash subseteq\; T;$ the argument for when $T\; \backslash subseteq\; S$ is similar. Then $(x,\; y)\; \backslash in\; T.$ Since all relations produced by our process are transitive, $(x,\; z)$ is in $T$ and therefore also in $\backslash bigcup\; \backslash mathcal.$ Similarly, it can be shown that $\backslash bigcup\; \backslash mathcal$ is antisymmetric.
Therefore by Zorn's lemma the set of partial orders containing $R$ has a maximal element $Q,$ and it remains only to show that $Q$ is total. Indeed if $Q$ had a pair of incomparable elements then it is possible to apply the process of the first step to it, leading to another strict partial order that contains $R$ and strictly contains $Q,$ contradicting that $Q$ is maximal. $Q$ is therefore a total order containing $R,$ completing the proof.
Other extension theorems

Arrow stated that everypreorder
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

(reflexive and transitive relation) can be extended to a total preorder
The 13 possible strict weak orderings on a set of three elements . The only total orders are shown in black. Two orderings are connected by an edge if they differ by a single dichotomy.
In mathematics
Mathematics (from Ancient Greek, Gre ...

(transitive and connex relation), and 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 $xRy$ for every pair of consecutive elements $(x,\; y),$ and there is some pair of consecutive elements $(x,\; y)$ in the cycle for which $yRx$ does not hold.
See also

*References

{{reflist Axiom of choice Theorems in the foundations of mathematics Articles containing proofs Order theory