Szpilrajn extension theorem

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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 Kuratowski–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

A
binary 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 $\left(x, y\right)$ of elements of $X,$ and $\left(x, y\right) \in R$ is often abbreviated as $xRy.$ A relation is reflexive if $xRx$ holds for every element $x \in X;$ 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 $xRy \text yRz$ imply $xRz$ for all $x, y, z \in X;$ it is antisymmetric if $xRy \text yRx$ imply $x = y$ for all $x, y \in X;$ 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 $xRy \text yRx$ holds for all $x, y \in X.$ 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 $x, y \in X.$ 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 $\left(x, y\right)$ and then performing the
transitive 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 $qRx \text yRp.$ 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 Kuratowski–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 $\mathcal$ be such a chain, and it remains to show that the union of its elements, $\bigcup \mathcal,$ is an upper bound for $\mathcal$ which is in the poset: $\mathcal$ contains the original relation $R$ since every element of $\mathcal$ is a partial order containing $R.$ Next, it is shown that $\bigcup \mathcal$ is a transitive relation. Suppose that $\left(x, y\right)$ and $\left(y, z\right)$ are in $\bigcup \mathcal,$ so that there exist $S, T \in \mathcal$ such that $\left(x, y\right) \in S \text \left(y, z\right) \in T.$ Since $\mathcal$ is a chain, either $S \subseteq T \text T \subseteq S.$ Suppose $S \subseteq T;$ the argument for when $T \subseteq S$ is similar. Then $\left(x, y\right) \in T.$ Since all relations produced by our process are transitive, $\left(x, z\right)$ is in $T$ and therefore also in $\bigcup \mathcal.$ Similarly, it can be shown that $\bigcup \mathcal$ 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 every
preorder 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 $\left(x, y\right),$ and there is some pair of consecutive elements $\left(x, y\right)$ in the cycle for which $yRx$ does not hold.