extension of a poset
Let $(P,{\le}_{P})$ be a poset. An extension^{} of $(P,{\le}_{P})$ is a poset $(Q,{\le}_{Q})$ such that

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$Q=P$, and

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if $a{\le}_{P}b$, then $a{\le}_{Q}b$.
From the definition, the underlying set of an extension of a poset does not change. We may therefore, when there is no ambiguity, say that $Q$ is an extension of a poset $P$ and use ${\le}_{Q}$ and ${\le}_{P}$ to distinguish the partial ordering on $Q$ and $P$ respectively.
Every poset has a trivial extension, namely itself. A poset is maximal if the only extension is the trivial one. Given a poset that is not maximal, can we find a nontrivial extension? Suppose $P$ is a poset and $(a,b)\notin {\le}_{P}$ ($a,b$ not comparable^{}). Then ${\le}_{Q}:={\le}_{P}\cup A$, where $A=\{(r,s)\mid r\le a\text{and}b\le s\}$, is a nontrivial partial order extending ${\le}_{P}$, nontrivial since $(a,b)\in {\le}_{Q}{\le}_{P}$. So a maximal poset is just a linearly ordered set, as any pair of elements is comparable.
An extension $Q$ of $P$ is said to be linear if $Q$ is a linearly ordered set. By Zorn’s lemma, and the construction above, every poset $P$ has a linear extension: take $\mathcal{P}$ to be the poset of all extension of $P$ (ordered by inclusion), given a chain $\mathcal{C}$ of elements in $\mathcal{P}$, the union $\bigcup \mathcal{C}$ is an element of $\mathcal{P}$, so that $\mathcal{P}$ has a maximal element^{}, which can easily be seen to be linear. Thus we have just proved what is known as the order extension principle:
Proposition 1 (order extension principle)
Every partial ordering on a set can be extended to a linear ordering.
And since every set is trivially a poset, where $a\le b$ iff $a=b$, we record the following corollary, known as the ordering principle:
Corollary 1 (ordering principle)
Every set can be linearly ordered.
In fact, every poset is the intersection^{} of all of its linear extensions
$$P=\bigcap \{L\mid L\text{is a linear extension of}P\},$$ 
for if $(a,b)$ is not in the intersection, a linear extension $M$ of $P$ can be constructed via above containing $(a,b)$, which is a contradiction^{}. A set $\mathcal{S}$ of linear extensions of a poset $P$ is said to be a realizer of $P$ if $\bigcap \mathcal{S}}=P$. Every poset has a realizer.
Remark. Instead of the stronger axiom of choice^{}, the order extension principle can be proved using the weaker Boolean prime ideal theorem. Furthermore, the ordering principle implies the axiom of choice for finite sets.
References
 1 W. T. Trotter, Combinatorics and Partially Ordered Sets, JohnsHopkins University Press, Baltimore (1992).
 2 T. J. Jech, The Axiom of Choice, NorthHolland Pub. Co., Amsterdam, (1973).
Title  extension of a poset 
Canonical name  ExtensionOfAPoset 
Date of creation  20130322 16:31:57 
Last modified on  20130322 16:31:57 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  15 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06A06 
Synonym  linear extension 
Defines  maximal poset 
Defines  linear extension of a poset 
Defines  realizer 
Defines  ordering principle 
Defines  order extension principle 